TitleAn introduction to Ocneanu's theory of double triangle algebrasfor subfactors and classification of irreducible connections onthe Dynkin diagrams (Recent Topics in Operator Algebras)

Author(s) Goto, Satoshi

Citation 数理解析研究所講究録 (1999), 1077: 105-139

Issue Date 1999-02

URL http://hdl.handle.net/2433/62644

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

An introduction to Ocneanu’s theory of double triangle algebrasfor subfactors and classification of $\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{C}\mathrm{i}‘ \mathrm{b}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{o},\mathrm{n}$nections on

the Dynkin diagrams

SATOSHI GOTO (SOPHIA UNIVERSITY)後藤聡史 (上智大学)

January 19, 1999

1 Introduction

The classification of subfactors of the hyperfinite $\mathrm{I}\mathrm{I}_{1}$ factor is one of the most importantand stimulating problems in the theory of operator algebras since V. F. R. Jones initiatedhis celebrated index theory for subfactors in [16]. The strongest form of the classificationhas been obtained by S. Popa based on his notion of strong amenability in [26]. In theearly stage it was known that hyperfinite $\mathrm{I}\mathrm{I}_{1}$ subfactors with index less than four has oneof the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6},$ $E_{7}$ and $E_{8}$ as their principal graphs. For the completeclassification of subfactors of the hyperfinite $\mathrm{I}\mathrm{I}_{1}$ factor A. Ocneanu introduced the notionof paragroup in [20]. And by the paragroup theory the classification of subfactors of thehyperfinite $\mathrm{I}\mathrm{I}_{1}$ factor with finite index and finite depth is reduced to the classification offlat $bi$-unitary $co\mathrm{n}$nections on the (dual) principal graphs.

The paragroup theory as well as its importance are now widely spread and moreand more people have become to work on the theory. The importance of paragrouptheory is not only because it is complete invariant for subfactors of the hyperfinite $\mathrm{I}\mathrm{I}_{1}$

factor with finite index and finite depth but also because it has deep relations to manyother theories in mathematics and mathematical physics. Actually it has been revealedthat there are striking relations between the paragroup theory and other theories suchas exactly solvable integrable lattice nuodels, quantun] groups, topological quantum fieldtheories (TQFT) both in the sense of Turaev-Viro ([32]) based on triangulation and in thesense of Reshetikhin-Turaev ([27]) based on surgery, $\mathrm{a}\mathrm{n}_{1}\mathrm{d}$ rational conformal field theories(RCFT) in the sense of Moore-Seiberg ([19]) and so on. (See for example [6], [11], [9],[10], [22], [23], [28], [33], [34]. All of these relations are explained in [13].)

In 1995 A. Ocneanu gave a series of lectures $011$ subfactor theory at The Fields Institutefrom April 19 to 25. In his lectures ([24]) he introduced a new algebra called $\mathrm{d}o\mathrm{u}ble$

triangle algebra by using the notion of essential paths and extension of Kauffman-Lins’Temperley-Lieb recoupling theory. He also gave many applications of his result. AInongtheir applications he raised particularly five problems in his talks at Aarhus in June 1995,

which consists of one problem concerned with TQFT, one concerned with RCFT andthree concerned with subfactor theory. There he showed that his method gives essentiallyone solution of them.

Among the solutions of the five problems the most fundamental result is the completeclassification of irreducible $\mathrm{b}\mathrm{i}$-unitary connections on the Dynkin diagrams An’ $Dn’ E6,7,8$

and other solutions will follow from it. More precisely the irreducible connections on theDynkin diagrams here means the irreducible connections on the four graphs which have the

数理解析研究所講究録1077巻 1999年 105-139 105

Dynkin diagram $I\mathrm{i}’$ and $L$ as the two horizontal graphs. (We call such a connection a K-L$\mathrm{b}i$-unit$a\mathrm{r}y$ connection.) And the classification $\mathrm{h}\mathrm{e}\mathrm{l}\cdot \mathrm{e}1\coprod \mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}$ the classification of irreducible

$\mathrm{b}\mathrm{i}$-unitary connections up to$\mathrm{g}e$auge choice, which is finer than the classificatioll up to

isomorphisms.The main purpose of this paper is to give a detailed proof of the classification of

irreducible connections on the Dynkin diagranls. As we mentioned in the beginningthe classification in more restricted case when the four graphs are all the same Dynkindiagrams has been done in order to classify subfactors with index less than four. So theclassification of connections itself is very important for this purpose.

Another example in which $\mathrm{b}\mathrm{i}$-unitary connections on the Dynkin diagrams naturallyappear is the construction of a series of subfactors given by Goodman-de la Harpe-Jones([14]). These subfactors are called Goodman-de la $H_{\dot{\mathrm{c}}}t1pe$-Jones $\mathrm{s}u$ bfactors. (We call themGHJ subfactors in short. A. Ocneanu calls the salne subfactors Jones-Okam$oto$ subfactorsbecause S. Okamoto computed their principal graphs [25].) They are constructed fromA-K $\mathrm{b}\mathrm{i}$-unitary connections, where $A$ represents the Dynkin diagrams $A_{n}$ and $I\iota’$ is one ofthe A-D-E Dynkin diagrams. The principal $\mathrm{g}\mathrm{r}\mathrm{a}_{1}$) $\mathrm{h}\mathrm{S}$ of these subfactors are easily obtainedby a simple method but the dual principal graphs as well as their fusion rules are muchmore difficult to compute. The most important example is the subfactor with index $3+\sqrt{3}$

which is constructed from the embedding of the string algebra of $A_{11}$ to that of $E_{6}$ , i.e., it$\mathrm{i}\mathrm{s}_{l}$ obtained from an $A_{11}- E_{6}\mathrm{b}\mathrm{i}$ -unitary connection. In this particular case it happens that itis not very difficult to compute the dual principal graph (see [18], [13, Section 11.6]). Butit is more difficult to determine its fusion rule. Actually D. Bisch has tried to compute thefusion rule just from the graph but there were five possibilities and it turned out that the$\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}_{\mathrm{o}\mathrm{n}}$ rule can not be determined $\mathrm{f}\mathrm{r}\mathrm{o}\ln$ the graph only. Some more information is neededand Y. Kawahigashi obtained the fusion rule as an application of paragroup actions in[18]. In his lectures at The Fields Institute A. Ocneanu gave a solution to this problem ofdetermining the dual principal graphs and their fusion rules as one of some applicationsof his theory of double triangle algebra ([24]). In particular, the fusion rule algebra of allK-K $\mathrm{b}\mathrm{i}$-unitary connection is used to determine the fusion rule of GHJ subfactors whichcorrespond to the Dynkin diagram $I\iota’$ . Here we would like to mention that some of recentworks has revealed a surprising relation between GHJ subfactors and conformal inclusions([37], [2], [3], [4]). Furthermore some generalization of the construction of GHJ subfactorshas been obtained by F. Xu $([3\dot{5}], [36])$ and J. B\"ockenhauer-D. E. Evans ([2], [3], [4]).

Another striking and unexpected observation which A. Ocneanu has found ([24]) isthe relation between fusion rule algebras of all K-K connections for a Dynkin diagram$K$ he obtained and affine $\mathrm{S}\mathrm{U}(2)$ modular invariants corresponding to the graph $I\mathrm{c}^{\nearrow}$ . TheA-D-E classification of affine $\mathrm{S}\mathrm{U}(2)_{1}\mathrm{n}\mathrm{o}\mathrm{d}_{\mathrm{U}}1\mathrm{a}\mathrm{r}$ invariants has been obtained in [7], (see also[31] $)$ . A. Ocneanu showed some interpretations of off-diagonal terms of these modularinvariant matrices corresponding to $D_{n}$ and $E_{6,7,8}$ in his lectures [24] by using a notionof essential paths. In December 1997, he introduced the notion of quantum $I_{1^{r}}l6inia\mathrm{n}$

invariants which is the quantum version of Kleinian invariant and he showed another newexplanation of the off-diagonal terms. After A. Ocneanu’s work on the fusion rule algebrasof K-K $\mathrm{b}\mathrm{i}$-unitary connections essentially the same fusion rule algebras are constructedfrom conformal inclusions of $\mathrm{S}\mathrm{U}(2)$ $\mathrm{W}\mathrm{e}\mathrm{s}\mathrm{S}^{-}\mathrm{z}\mathrm{u}\min_{0}$ -Witten models by F. Xu ([37]) and J.B\"ockenhauer-D. E. Evans ([3], [4]) in flat cases. Note that the non-flat case, i.e., thecase of $D_{odd}$ and $E_{7}$ can not be obtained fron] their approach using conformal inclusions.Moreover the theory of double triangle algebra is recently used to generalize the result tothe case of conformal inclusions of $\mathrm{S}\mathrm{U}(7t)$ WZW models by J. B\"oCkenhauer-..D. E. Evans-Y.

106

Kawahigashi ([5]). They showed that A. Ocneanu’s observation of the relation between

fusion rule algebras and modular invariant matrices holds true for some more general

cases including the case of $\mathrm{S}\mathrm{U}(n)$ WZW models.Now we give a brief outline of the contents in this paper. In the next section we will

give some definitions and terminologies concerning A. Ocneanu’s double triangle algebras

and we also fix some notations. Though all of the definitions of important notions such

as essential paths, gaps of finite graphs and chiral projectors are given in [24], we did not

omit them for reader’s convenience because they are indispensable for the classificationof connections. We refer readers to [24] for more details.

In section 3 we define some operations on the set of connections such as direct sum,

conjugation, irreducible decomposition and composition (product). These operations arefirst defined by A. Ocneanu in [24] and later M. Asaeda-U. Haagerup clarified the cor-respondence between these operations on connections and those on bimodules ([1]). In

order to deal with the system of connections closed under these operations we define

a notion of horizontally conjugate pair of connections and give a natural identificationbetween connections. We also give sonle equivalence relations on connections. We will

emphasize the difference of $ve\mathrm{r}$tical $g\mathrm{a}u\mathrm{g}e$ choice and total gauge choice and will make it

clear that the $v\mathrm{e}\mathrm{r}$tical gauge choice is the right equivalence relation to deal with a system

of connections. This point is also clarified by M. Asaeda-U. Haagerup ([1]). In order to

make the most of M. Asaeda-U. Haagerup’s notion of generalized open string bimod$\mathrm{u}le$

we will show that Frobenius reciprocity holds for the $\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\ln$ of connections.Section 4 is devoted to show the correspondence between irreducible K-L bi-unitary

connections and $\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{C}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}*$ -representations of the double triangle algebras on the graphs$K$ and $L$ . This is the most important tool to classify the irreducible connections. Though

a detailed proof of the correspondence is given in his original paper [24], some more details

are necessary for our purpose. So we will supply it here and as a corollary we show animportant correspondence between $\mathrm{s}\mathrm{o}\mathrm{n}$) $\mathrm{e}$ special lllinimal central projections of the double

triangle algebra and some irreducible $\mathrm{b}\mathrm{i}$-unitary connections on the Dynkin diagralns. We

also show the relation between the fusion rule algebra of K-K $\mathrm{b}\mathrm{i}$-unitary connectioroe and

the center of the double triangle algebra. Actually it turns out that the fusion rule algebra

is isomorphic to the center of double triangle algebra with different product from original

one. One will notice that the notion of horizontally conjugate pair and the equivalence

relation $ve\mathrm{r}$tical $ga\mathrm{u}ge$ choice are both natural to deal with this correspondence.Finally in section 5, the classification result is explained in each case of the Dynkin

diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ . In the procedure to get this result we also obtain the new fusion

rule algebras which consists of all K-K $\mathrm{b}\mathrm{i}$-unitary connections. It also provides a simple

proof of the flatness of $D_{2n},$ $E_{6}$ and $E_{8}$ connections. Hence we get another proof of the

complete classification of subfactors of the hyperfinite $\mathrm{I}\mathrm{I}_{1}$ factor with index less than 4 by

this method. The flat part of non-fiat connections $D_{2n+1}$ and $E_{7}$ are also obtained easily.

By putting together all the cases of $A- D_{- E}$ we will obtain some important structural

result on the fusion rule algebras including a partial $\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{V}}\mathrm{i}\mathrm{t}\mathrm{y}$ of the fusion rule

algebra.

2 Preliminaries and Notations

In this section we give definitions of essential $\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{S}$ on finite graphs and the double triangle

algebras for the sake of completeness. We also fix some notations. We refer readers to

[24] for the details.

107

$\mathrm{r}n$

$c$ $=$

A $\mathit{1}\mathit{1}B$ $B^{\Pi}A$

Figure 1: Creation operator

$c^{*}$ $=$

$v$

Figure 2: Annihilation operator

2.1. $We\mathrm{n}zl$ projectors and Essential path$\mathrm{s}$ . $\mathrm{C}^{\mathrm{t}}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r}$ a finite oriented bipartite graph$\mathcal{G}$ . We denote even vertices of the graph $\mathcal{G}$ by $\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}\mathcal{G}0$ and its odd vertices by $\mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{1}}\mathcal{G}$ .We denote an abelian $C^{*}$-algebra with basis $u\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}^{0}\mathcal{G}$ [resp. $v\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}^{1}\mathcal{G}$ ] by $A$ [resp.$B]$ . We use a notation $H$ for a Hilbert space with basis $e\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}^{(0,1}$

)$\mathcal{G}$ . Here Edge $(0,1)\mathcal{G}$

represents the set of oriented edges of $\mathcal{G}$ with orientation from $\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}^{0}\mathcal{G}$ to $\mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{1}}\mathcal{G}$ . TheHilbert space $H$ defined as above bec.olnes an A-B binlodule with the action defined asfollows, $x\cdot\xi\cdot y=\delta_{x,S(}\delta\xi$) $r(\xi),y\epsilon$ for $x\in \mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{0}}\mathcal{G},$ $y\in \mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{1}}\mathcal{G}$ and $\xi\in$ Edge $(0,1)\mathcal{G}$ . Here$s(\xi)$ [resp. $r(\xi)$ ] means the source (starting point) [resp. range (end point)] of the edge$\xi$ . The Hilbert space $H$ has the conjugate $\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{l}\cdot \mathrm{t}$ space $\overline{H}$ with the basis consists of theorientation reversed edges $\overline{\xi}\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}^{(1,0}$

)$\mathcal{G}$ . Then the conjugate Hilbert space $\overline{H}$ becomes

a B-A bimodule. We denote the adjacency matrix of $\mathcal{G}$ by $\triangle_{\mathcal{G}}$ and the Perron-Frobeniuseigenvector for $\triangle_{\mathcal{G}}$ by $\mu$ which satisfy the condition $\triangle_{Q\mu}=\beta\mu$ . Here $\beta$ is the Perron-Frobenius eigenvalue of $\triangle_{\mathcal{G}}$ .

Now we define the annihilation operator $c\in \mathrm{H}_{0111}(AHB\otimes{}_{B}\overline{H}_{A}, A)$ by the following.

$c(\xi\otimes\overline{\eta})=\delta_{\xi,\eta}\mu(\gamma(\xi))^{1/}2\mu(S(\xi))-1/2$ . $s(\xi)$

for $\xi,$ $\eta\in$ Edge $\mathcal{G}$ .Its adjoint operator is called the creation operator which is given by the following.

$c^{*}(_{X)}= \sum_{=\xi\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}Q,s(\xi)x}\xi\otimes\overline{\xi}\mu(r(\xi))^{1/}2(\mu S(\xi))^{-1/2}$ .

We draw diagrams for the operators $c$ and $c^{*}$ as in Figures 1 and 2.We remark that the composition of these operators $c\cdot c^{*}$ becomes scalar multiplication

operator $\beta$ . We define the Jones projection on the path spaces on the graph $\mathcal{G}$ by $e=$$\beta^{-1}c^{*}\cdot c$ . For these operators we draw pictures as in Figures 3 and 4.

$\Pi|$

$c\cdot c^{*}=$ $=\beta$

$w$

Figure 3:

108

$H$ $\overline{LT}$

$e=\beta^{-1}c^{*}\cdot c=\beta^{-1}$

$Il$ $t\mathrm{f}$

Figure 4: The Jones projection

$p_{n}=1-e_{1}\vee e_{2}$

.

$\mathrm{v}\cdots\vee en-1$ $=$ $=$ $|n$

Figure 5: Wenzl projector $p_{n}$

We denote the set of paths of a graph $\mathcal{G}$ with length $n$ by Path $(n)\mathcal{G}$ , i.e. Path$()n\mathcal{G}=$

{ $\xi=(\xi_{1},$ $\xi_{2},$$\cdots,$ $\xi_{n})|\xi_{k}\in$ Edge $\mathcal{G},$ $s(\xi k+1)=r(\xi_{\iota)}.$ }, and denote the Hilbert space

with orthonormal basis $\xi\in \mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}^{()}n\mathcal{G}$ by $\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$ )$\mathcal{G}$ . Note that $n$ times (relative) tensor

products ${}_{A}H_{B}\otimes{}_{B}\overline{H}_{A}\otimes\cdots\otimes {}_{A}H_{B}(\mathrm{o}\mathrm{r}_{B}\overline{H}_{A})$ produce the path Hilbert space $\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$ )$\mathcal{G}$ .

We can define the sequence of $\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}/\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{i}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ operators $c_{1},$ $c_{2,.n-1},.C$ alld that ofthe Jones projections $e_{1},$ $e_{2},$ $\ldots e_{n-1}$ on the Hilbert space $\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$ )

$\mathcal{G}$ depending on theposition where they act.

Definition 2.1 The Wenzl projectors $p_{n}$ on $\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$ )$\mathcal{G}$ is defined by $p_{n}=1-e_{1}\vee e_{2}\vee$

... V $e_{n-1}.\dot{\mathrm{W}}\mathrm{e}$ draw the picture in Figure 5 for the $\dot{\mathrm{W}}$enzl projector $p_{n}$ . The space ofessential $p\mathrm{a}ths$ with length $n$ on a graph $\mathcal{G}$ is defined by $\mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}(n)\mathcal{G}=p_{n}\cdot \mathrm{H}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)\mathcal{G}$ .We denote the space of essential paths of a graph $\mathcal{G}$ with length $n,\acute{\mathrm{w}}\mathrm{i}\mathrm{t}\mathrm{h}$ starting point $x$

and end point $y$ by $\mathrm{E}\mathrm{s}\mathrm{s}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}_{x}(n,)\mathcal{G}y$ .

We remark that the space of essential paths can be defined as follows.

$\mathrm{E}\mathrm{s}\mathrm{s}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)\mathcal{G}$ – $\{\xi\in \mathrm{H}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}^{(n})\mathcal{G}|e_{k}\xi=0\mathrm{f}\mathrm{o}1k=1,2, , \cdots, n-1\}$

$=$ { $\xi\in \mathrm{H}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)\mathcal{G}|c_{k}\xi=0$ for $k=1,2,$$,$

$\cdots$ , $n-1$ }.

The following Moderated Pascal rule is quite useful to count a dimension of essentialpaths.

$\dim \mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}^{(n+)}a,x1\mathcal{G}=\xi\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}\mathcal{G}\sum_{\xi r()=x},$

dinl $\mathrm{E}\mathrm{s}\mathrm{S}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}_{a,S(\xi}(n)\mathcal{G}-\dim \mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}(n-1)\mathcal{G})a,x$

For the proof of this rule, see [24, Section 5].2.2. Extension of recoupling model and the double triangle algebra. Next we define an

extended model of Kauffman-Lins’ recoupling theory ([17]) from a viewpoint of subfactortheory by using the notion of essential paths. First we remark that the recoupling $\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}$

for $q=e^{i\pi/N}$ a root of unity can be $1^{\cdot}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{i}_{\mathrm{Z}\mathrm{e}}\mathrm{d}$ by using the fusion rule algebra of sector(or bimodule) and quantum $6\mathrm{j}$-synlbols arising fionl the Jones’ subfactor with principal

109

$\in \mathrm{H}_{0}\mathrm{m}(\sigma_{n},\otimes\sigma n’\sigma_{k})$ .

Figure 6: An intertwiner

$b$

$d$

$\mathrm{F}\mathrm{i}\mathrm{g}\iota 11^{\backslash }\mathrm{e}7:\mathrm{R}^{1}‘ \mathrm{c}()\iota 11)1\mathrm{i}_{1}$

graph $A_{N-1}$ . For example the trivalent $\mathrm{v}\mathrm{e}\mathrm{l}\cdot \mathrm{t}\mathrm{e}\mathrm{x}$ as in $\mathrm{F}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{l}\cdot \mathrm{e}6$ represent an intertwinerin $\mathrm{H}\mathrm{o}\mathrm{m}(\sigma_{m}\otimes\sigma_{n}, \sigma_{k})$ . Here $\sigma_{j}$ is an irreducible sector (or bimodule) $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{0}11\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$ toj-th vertex $\mathrm{f}\mathrm{r}\mathrm{o}\ln$ the distinguished $\mathrm{v}\mathrm{e}\mathrm{l}\cdot \mathrm{t}\mathrm{e}\mathrm{x}*\mathrm{o}\mathrm{f}$ the prillcipal graph of type $A_{N-1}$ Jones’subfactors. The other notions in the recoupling theory stlch as $\theta$-evaluations, tetrahedralnets and (quantum) $6\mathrm{j}$-symbols will be $\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{l}\backslash \mathrm{e}\mathrm{t}\mathrm{e}\mathrm{d}$ in ternls of sectors and intertwinersarising from the Jones’ subfactor. Especially we llave the recoupling as in Figure 7. Herethe coefficient is given as follows.

$( \mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f})_{n}==\frac{\mathrm{T}\mathrm{e}\mathrm{t}\{\begin{array}{lll}Cl b nc d m\end{array}\}\triangle_{n}}{\theta((l,b_{\mathrm{t}}/\mathrm{t})\theta(_{C,d,n)}}$

Here $\theta(a, b, c)$ means the $\theta$-evaluation and

Tet

represents a value of the tetrahedral net. (See [17], [24, Section 12].) The special casewhen $m=0$ is given in Figure 8 and we use this to define the convolution product of thedouble triangle algebras.

Now fix a recoupling model $A$ which corresponds to a Perron-Frobenius eigenvalue $\beta$

and let $K$ be one of the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ with the same Perron-Frobeniuseigenvalue. We draw a picture for an essential path $\xi\in \mathrm{E}\mathrm{s}\mathrm{s}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)Kx,y$ as in the left hand

”. $h$

$|$ $|$

$=$ $\sum_{n}\frac{\triangle_{n}}{\theta(a,b,n)}$

a $b$ $o$

Figure 8:

110

$v$

Figure 9:

Figure 10:

side of Figure 9 or we simply draw the picture in the right hand side. (See [24, Section10].) Then a double triangle algebra $A$ is defined as an algebra which elements are linear

combinations of pairs of essential paths as in Figure 10. We call a double triangle algebradefined on the graphs $K$ and $L$ which correspond to upper and lower horizontal graphsrespectively a K-L double triangle alge $\mathrm{b}ra$ . Two products are defined on this algebra.

One is. product defined as in Figure 11. The other product called convolution prod$uct$

is defined as in Figure 12 and denoted $\mathrm{b}\mathrm{y}*$ . We $\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{P}\mathrm{o}\mathrm{s}\mathrm{e}$ the element of the right handside in this figure as in Figure 13 by using recoupling and the equality in Figure 14. The$*$-operation for the convolution product on the double triangle algebra is given by Figure

15.2.3 Ocneanu’s chiral projectors. We recall that special elements in $(A, *)$ is defined by

$\delta_{\xi_{-},\eta+}$

Figure 11:. product on the double triangle algebra

111

Figure 12: The convolution product on the double triangle algebra

Figure 13:

Figure 16 and Figure 17 and give the definition of Ocneanu’s chiral projectors.

Definition 2.2 The chiral projectors $\Psi_{+}$ and $\Psi$ -which are central projections in thedouble triangle algebra $(A, *)$ are defined as in Figure 18 and 19. The product of the twochiral projectors $\Psi_{+}*\Psi_{-}$ is called the ambichiral projector and is denoted by $\Psi_{\pm}$ .

2.4 Gaps on the Dynkin diagrams and $\mathrm{m}$ inim$\mathrm{a}l$ central projections. The gap and O-gapof a finite graph $G$ are numbers (positive integer or $\infty$ ) defined by the following. (See [24,Section 17])

gap$(c)$ $\equiv$ $\min$ { $n>0|\mathrm{E}\mathrm{s}\mathrm{s}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}^{(}|l)Ga,a\neq 0$ for all $a\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}G$},$\mathrm{O}-\mathrm{g}\mathrm{a}\mathrm{p}(c)$ $\equiv$ $\min\{n>0|\mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(n)}0,0G\neq 0\}$.

Here $0$ represents the distinguished vertex of the $\mathrm{C}^{\mathrm{t}}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}$ graph $G$ , i.e., the vertex of $G$

which has the smallest Perron-Frobenius weight. The gaps and the $0$-gaps of the Dynkindiagrams are given in the following table.

Figure 14:

112

$=( \frac{\mu(y)\mu(_{\sim}^{\sim})}{\mu(x)\mu(lv)})^{1/2}$

Figure 15: The $*-\mathrm{o}\mathrm{p}\mathrm{e}1^{\cdot}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}\mathrm{f}\mathrm{o}\mathrm{l}\backslash \mathrm{t}\mathrm{h}\dot{\mathrm{e}}$convolution product$x$ $\alpha$ $y$

$\equiv\sum_{x,y,\alpha}\overline{\mathrm{o}}_{x}^{1/2}\overline{\mathrm{O}}_{y\uparrow x}^{1}/2\mathrm{O}$

$x$ $\overline{\alpha}$ $y$

Figure 16:

Let $K$ be a connected finite bipartite graph and $(A, *)$ a double triangle algebra on $K$

endowed with the convolution product. The following proposition shows that there aretwo finite family of minimal central projections $\{p_{k}^{+}$. $\}_{k}$ and $\{p_{k}^{-}$. $\}_{k}$ on the double trianglealgebra $(A, *)$ .

Proposition 2.3 ([24, Proposition 17.3, Corollary 17.4]) Two $element\mathit{8}p_{k}+$ and $p_{k}^{-}$ definedby Figure 20 and 21 are minimal central projections if $k<\mathrm{g}\mathrm{a}\mathrm{p}(K)/2$ .

For orthogonality of these projections we have the following propositions.

Proposition 2.4 ([24, Proposition 18.1]) The $\gamma’$? inimal central projections $p_{k}^{+}$ (resp. $p_{k}^{-}$ )are mutually orthogonal if $k<\mathrm{g}\mathrm{a}\mathrm{p}(I_{1}’)/2$ .

$x$ $\alpha$ $y$

$\equiv\sum_{x,y,n\alpha},\overline{\mathrm{O}}_{xJ}1/2\overline{\mathrm{o}}_{1}^{1}\mathrm{O}_{7}/2l$

$x$ $\overline{\alpha}$ $y$

$\mathrm{F}\mathrm{i}\mathrm{g}\iota\iota \mathrm{r}\mathrm{e}17$ :

113

Figure 18:

Figure 19:

Proposition 2.5 ([24, Proposition 18.2]) The minimal central projections $p_{k}^{+}$ and $p_{l}^{-}$ asin the previous section are mutually orthogonal if $k\neq l$ and $k+l<0- \mathrm{g}\mathrm{a}\mathrm{p}(K)$ .

3 Operations and equivalence relations of connections, conjugate pairs andAsaeda-Haagerup’s generalized open string bimodules

In this section we define some operations on the set of connections such as direct sum,composition, irreducible decomposition and conjugation etc. (See [24, Section 20].) Wealso define some equivalence rela.tions OI1 it. 1Ve sllall give a natural identification ofconnections and define the above operations on the set of equivalence classes of connectionswith the identification. These operatiolls are originally defined by Ocneanu ([24, Section20]). Later Asaeda and Haagerup ([1]) $\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{l}\cdot \mathrm{o}\mathrm{d}\iota \mathrm{t}\mathrm{C}\mathrm{e}\mathrm{d}$ the uotion of generalized open stringbimodules which is a $\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}_{0}11$ of open $\mathrm{s}\mathrm{t}\mathrm{l}\cdot \mathrm{i}\mathrm{l}$ bimodule of Ocneanu ([20]) alld Sato([30]) and they clarified the relation between connections and bimodules. We rernarkthat the identification of connections given in this section is different Asaeda-Haagerup’ssetting. Here we will also give their original definitions for reader’s convenience.

Remark 3.1 In this paper connections on four graphs $\mathcal{G}0,$ $\mathcal{G}_{1},$ $\mathcal{G}2,$ $\mathcal{G}_{3}$ as in Figure 22 arealways assumed to have connected horizontal graphs $\mathcal{G}_{0}$ and $\mathcal{G}_{2}$ . We do not $\mathrm{a}\mathrm{s}\mathrm{s}\iota \mathrm{l}\mathrm{m}\mathrm{e}$ thatthe vertical graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{3}$ are connected. The word connection always means bi-unitaryconnection in this paper. So we will often use $\mathrm{t}1_{1}\mathrm{e}$ word connection instead of ‘ $bi_{-un}i$faryconnection’ for simplicity.

First we define the notion of direct sunl, $\mathrm{c}\mathrm{o}\mathrm{n}$) $\mathrm{P}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ , irreducibility and conjugation onthe set of $\mathrm{b}\mathrm{i}$-unitary connections. (See [24, Section 20], [1, Section 3].)

$p_{k}^{+}\equiv$ $\leq\Psi_{+}$

Figure 20:

114

$p_{k}^{-}\equiv$$\leq\Psi_{-}$

Figure 21:$\mathcal{G}_{0}$

$\mathcal{G}_{3}$ $\mathcal{G}_{1}$

$\mathcal{G}_{2}$

Figure 22:

Definition 3.2 ([24, Definition 20.2]) Let $W_{1}$ and $W_{2}$ be two $\mathrm{b}\mathrm{i}$-unitary conllections onfour graphs $\mathcal{G}0,$ $\mathcal{G}_{1},$ $\mathcal{G}2,$ $\mathcal{G}_{3}$ and $\mathcal{G}_{0},$ $\mathcal{G}_{1’ 2}’\mathcal{G},$ $\mathcal{G}’3\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{c}}}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}1}\mathrm{y}$, then a direct sum of these bi-unitary connections is a $\mathrm{b}\mathrm{i}$-unitary connection $W$ on four graphs $\mathcal{G}_{0},$ $\mathcal{G}_{1}\mathrm{u}\mathcal{G}_{1}’,$ $\mathcal{G}2,$ $\mathcal{G}3\mathrm{u}\mathcal{G}_{3}’$

defined by the following. See Figure 23.

$W$ ($\xi_{3}\mathrm{i}_{\overline{\xi_{2}}}^{\underline{\xi}}.0$

.

$\downarrow..\xi_{1}$)$.$ .

$–$ $\{$

$W_{1}(\xi_{3}\mathrm{i}_{\overline{\xi_{2}}}^{\underline{\xi 0}}.\mathrm{i}.\xi 1)$

,

if $\xi_{0}\in \mathcal{G}_{0},$ $\xi_{1}\in \mathcal{G}_{1},$ $\xi_{2}\in \mathcal{G}2,$ $\xi 3\in \mathcal{G}_{3}$ ,

$W_{2}(\xi_{3}\mathrm{i}_{\overline{\xi_{2}}}^{\underline{\xi_{0}}}.\mathrm{i}.\xi 1)$

,

if $\xi_{0}\in \mathcal{G}_{0},$ $\xi 1\in \mathcal{G}_{1}’,$ $\xi_{2}\in \mathcal{G}_{2},$ $\xi_{3}\in \mathcal{G}_{3}/$ ,

$0$ , otherwise.

We denote a direct sum $\mathrm{b}\mathrm{i}$-unitary connection $W$ of two $\mathrm{b}\mathrm{i}$-unitary connections $W_{1}$ and$W_{2}$ by $W_{1}\oplus W_{2}$ .

Definition 3.3 ([24, Section 20]) Let $W$ be a connection on four graphs $\mathcal{G}0,$ $\mathcal{G}_{1},$ $\mathcal{G}2,$ $\mathcal{G}_{3}$

and $W’$ a connection on other four graphs $\mathcal{H}_{0}=\mathcal{G}_{2},$ $\mathcal{H}_{1},$ $\mathcal{H}_{2},$ $\mathcal{H}_{3}$ which has the conlmon

$\mathcal{G}_{0}$ $\cdot \mathcal{G}_{0}$$\mathcal{G}_{0}$

$\mathcal{G}_{3}$ $W_{1}$ $\mathcal{G}_{1}\oplus \mathcal{G}_{3}’$ $W_{2}$ $\mathcal{G}_{1}’$ $=\mathcal{G}_{3}\mathrm{u}\mathcal{G}_{3}’$ $W_{1}\oplus W_{2}$ $\mathcal{G}_{1}\mathrm{u}\mathcal{G}_{1}’$

$\mathcal{G}_{2}$$\mathcal{G}_{2}$

$\mathcal{G}_{2}$

Figure 23: Direct sum of two $\mathrm{b}\mathrm{i}$-unitary connections

115

$\mathcal{G}_{0}$

$\mathcal{G}’$ $W^{\prime/}$ $\mathcal{G}^{\prime/}$

$\mathcal{H}_{2}$

Figure 24: Composition (or product) of two connections

graph $\mathcal{H}_{0}=\mathcal{G}_{2}$ , then we define a composition of thelll by a $\mathrm{b}\mathrm{i}$-unitary connection $W^{\prime/}$

obtained by connecting these graphs, making products of both connections and summingthem over all the common horizontal edges as in Figure 24. Here the vertical graphs $\mathcal{G}’$

and $\mathcal{G}^{\prime/}$ of the composed connection $W^{\prime/}$ will change by this construction. We denote thecomposite connection $W^{\prime/}$ by $W\cdot W’$ or simply $WW’$ .

Definition 3.4 ([24, Definition 20.3]) A $\mathrm{b}\mathrm{i}$-unitary $\mathrm{c}\mathrm{o}\mathrm{n}11\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}W$ on four graphs $\mathcal{G}0,$ $\mathcal{G}_{1},$ $\mathcal{G}2,$ $\mathcal{G}_{3}$

are called reducible if there exist two $\mathrm{b}\mathrm{i}$-unitary connections $W_{1}$ and $W_{2}$ on four graphs$\mathcal{G}_{0},$ $\mathcal{G}_{1},$ $\mathcal{G}2,$ $\mathcal{G}_{3}$ and $\mathcal{G}0,$ $\mathcal{G}’1’ \mathcal{G}2,$ $\mathcal{G}’3\mathrm{P}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$such that the direct sum of them produces $W$

as in Figure 25 up to vertical gauge choice. We call a bi-unita.ry connection $W$ irreducibleif it is not reducible.

Remark 3.5 We remark that this definition of reducibility is the same as Asaeda-Haagerup’s([1, Section 3]). In general the reducibility up to vertical gauge choices is different fromthat up to total gauge choices. But the next $\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{a}$ shows that both definition coincidesif two horizontal graphs $\mathcal{G}_{0}$ and $\mathcal{G}_{2}$ as in Figure 22 are trees, i.e., if both graphs onlyhave single edges and have no cycle. (cf. [1, Relnarl\’’ Section 3].) Especially in the casewhen the horizontal graphs are both alllong the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ the twoequivalence relations coincide.

Lemma 3.6 If one of the two horizontal $graph_{6}\mathcal{G}_{0}w$ a tree, then any gauge choices on$\mathcal{G}_{0}$ can be forced to put on vertical gauge choice. In particular if the two horizontal $graph_{\mathit{8}}$

$\mathcal{G}_{0}$ and $\mathcal{G}_{2}$ are both trees, then two equivalence relation ‘total gauge choice’ and ‘uerticalgauge choice’ on the set of connections on the four graphs coincide.

Proof Take a gauge choice $\alpha\in \mathrm{C}(|\alpha|=1)$ on an edge of $\mathcal{G}_{0}$ . We can easily see thatthe same gauge choice can be given by taking a gauge choice on vertical graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{3}$

which consists only gauge $\alpha$ on some edges of $\mathcal{G}_{1}$ and $\alpha^{-1}$ on some edges on $\mathcal{G}_{3}$ . (This canbe shown by taking a vertical gauge step by step.) The existence of such a gauge

$\mathrm{c}\mathrm{h}_{0}\mathrm{i}\mathrm{C}\mathrm{e}\coprod$

is assured by the fact that there is no multiple edge and no cycle on the graph $\mathcal{G}_{0}$ .

Remark 3.7 The condition in the above lemma that the graph $\mathcal{G}_{0}$ does not have any cycleis necessary. Consider the $ca\mathit{8}e$ when all the four graphs are $A_{2n+1}^{(1}$

)$(n\geq 1)$ . It is easy to

see that you can not force to put horizontal gauge choice to vertical o.nes in these examples.$Thi\mathit{8}\mathit{8}hows$ that even if the two homzontal graphs consist of single edges, it may happenthat the above two equivalence $relation\mathit{8}$ do not coincide.

116

$\mathcal{G}_{2}$ $\mathcal{G}_{2}$ $\mathcal{G}_{2}$

$\mathcal{G}_{1}$ $W_{1}$ $\mathcal{G}_{3}\oplus \mathcal{G}_{1}$

;$W_{2}$ $\mathcal{G}_{3}’$ $=\mathcal{G}_{1}\mathrm{u}\mathcal{G}_{1}’$ $W$ $\mathcal{G}_{3}\mathrm{u}\mathcal{G}_{3}’$

$\mathcal{G}_{4}$ $\mathcal{G}_{4}$ $\mathcal{G}_{4}$

Figure 25: Reducibility of a $\mathrm{b}\mathrm{i}$-unitary connection $W$

$\mathcal{G}_{0}$ $\overline{\mathcal{G}}_{0}$

$\mathcal{G}_{3}$ $\mathcal{G}_{3}$

$\tilde{\mathcal{G}}_{3}$$\overline{\mathcal{G}}_{3}$

$\mathcal{G}_{0}$ $\tilde{\mathcal{G}}_{0}$

Figure 26: Four connections associated to a connection $W_{0}$

In this paper an equivalence relation on connections will mean vertical gauge choiceunless otherwise stated. Before we define a natural identification of connections we remarkfirst that four different connections which are transferred each other by renorlnalizationrule are associated to one $\mathrm{b}\mathrm{i}$-unitary connection $W_{0}$ as in Figure 26, where $\tilde{\mathcal{G}}$ representsthe graph $\mathcal{G}$ with reversed orientation.

Definition 3.8 We say two connections $W_{0}$ and $W_{1}$ (resp. $W_{0}$ and $W_{3}$ ) as in Figure 26are horizontally (resp. vertically) conjugate. We denote $\mathrm{t}\mathrm{h}\mathrm{e}\ln$ by $W_{0}^{h}$ and $W_{0}^{v}$ respectively.We will call vertically conjugate connection $W_{0}^{v}$ simply a conjugate connection of $W_{0}$ . Wedefine a conjugation operation on a set of connections by a vertical conjugation in thissense. The conjugation operation is often denoted by $-.$ . So we will also adopt the notation$\overline{W}$ for a (vertically) conjugate connection of $W$ .

In this paper, we regard two horizontally conjugate connections as the saIlue one. Inother words, we always consider a connection $W_{0}$ as a pair of two connections $(W_{0}, W_{1})$

which are horizontally conjugate each other. We call such a pair horizontally conjugatepair. By this identification the relation ‘ vertically conjugate’ still make sense. Moreover,we consider the two equivalence relation ‘ tot$\mathrm{a}lgau\mathrm{g}eCl1$oice’ and ‘

$ve\mathrm{r}$tical gauge choice’on this identified set of connections. Again they are still equivalence relation on it.

In the following we consider the set of connections which have fixed common horizontalgraphs. We will use the following terminology. (See [24, Section 20].)

Definition 3.9 Let $K$ and $L$ be two connected $\mathrm{f}\mathrm{i}_{11}\mathrm{i}\mathrm{t}\mathrm{e}$ bipartite graphs. A bi-unitaryconnection on four graphs is called a K-L $bi$-unitary connection if it has the graph $I\mathrm{i}^{\Gamma}$ asan upper horizontal graph and the graph $L$ as a lower horizontal graph as in Figure 27.

Note that we can naturally define the operations such as direct sum, conjugation andirreducible decomposition on the set of equivalellce classes of connections with the above

117

$I_{1}’$

$W$

$L$

Figure 27: K-L $\mathrm{b}\mathrm{i}$-unitary connection

$I_{1,*_{h}}^{\nearrow,I\mathrm{i}}ev\underline{enI_{1^{od}}^{\vee}\prime}d$

$I_{1^{od}}’d$$\overline{I1\prime}I\searrow’*_{f_{\mathrm{t}}}eve,n$

Figure 28: A graph $I\iota’$ and $\overline{I\iota^{r}}$

identification. Before we define the colluposition (or product) operation on it we needsome more notations and terminology.

We often denote a graph $K$ with a distinguished vertex $*_{K}$ by a pair $(I_{\mathrm{t}}’, *_{K})$ . Let$(I\mathrm{i}’, *_{K})$ be a connected finite bipartite graph with a distinguished vertex. A vertex of $I\mathrm{i}^{r}$

with the same (resp. different) colour as $*_{I\iota’}$ is called even (resp. odd) vertex. The setof even and odd vertices are denoted by $I\mathrm{i}\prime even$ and $I\iota\prime odd$ respectively. In the followingwhenever we consider a horizontal graph $(K, *_{I\iota’})$ , the notation $K$ will represent the graph$K$ with their even vertices on the left hand and odd vertices on the right hand. Thegraph $K$ with reversed orientation, that is, one with their odd vertices on the left andeven vertices on the right will be denoted by $\overline{\mathrm{A}’}$ . (See Figure 28.)

Note that when two horizontal graphs $I\iota^{r}$ and $L$ are connected finite bipartite withdistinguished vertices, four kinds of K-L $\mathrm{b}\mathrm{i}$-unitary connections will be distinguished bythe above notation which respects the orientations of the graphs. That is, there are fourkinds of K-L $\mathrm{b}\mathrm{i}$-unitary connections depending on which graph $I\mathrm{i}^{r}$ or $\overline{K}$ and $L$ or $\overline{L}$ theyactually have as horizontal graphs.

For a given connection we can naturally associate the index of its generalized openstring bimodule [1]. We call it an index of a connection. This value is the same as thesquare root of the index of subfactor constructed $\mathrm{f}\mathrm{r}\mathrm{o}\ln$ the connection.

Now we define a composition (or product) of two (pairs of) connections.

Definition 3.10 Let $(K, *_{K}),$ $(L, *_{L})$ and $(\mathit{1}\mathcal{V}I, *_{M})$ be three connected finite bipartitegraphs with distinguished vertices. Let $\alpha$ be a K-L $\mathrm{b}\mathrm{i}$ -unitary connection and $\beta$ a L-M$\mathrm{b}\mathrm{i}$-unitary connection. We regard these as two pairs of connections $(\alpha, \alpha^{h})$ and $(\beta, \beta^{h})$

as in the above setting. Define a $comp_{\mathit{0}}\mathrm{s}iti_{\mathit{0}}\mathit{1}1$ (or product) of two (pairs of) connections$\alpha$ and $\beta$ by a (pair of) connection obtained by the following procedure. When $\alpha$ and $\beta$

have the common graph $L$ as bottom and top $\mathrm{g}1^{\backslash }\mathrm{a}\mathrm{p}\mathrm{h}\mathrm{s}1^{\cdot}\mathrm{e}\mathrm{s}_{\mathrm{P}}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{e}\mathrm{l}\mathrm{y}$, then we just make aproduct $\alpha\cdot\beta$ and regard this as a $1$) $\mathrm{a}\mathrm{i}\mathrm{r}$ of connectiolls $(\alpha\cdot\beta, \alpha^{h}\cdot\beta^{h})$ . When one of $\alpha$ and$\beta$ have the graph $L$ as a top or $\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{l}\mathrm{l}\overline{\mathrm{l}}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}1_{1}$ alld tlle other have the graph $\tilde{L}$ , then wemake a product $\alpha\cdot\beta^{h}$ or a product $\alpha^{h}\cdot\beta$ . Again we regard it as a pair of connections$(\alpha\cdot\beta^{h}, \alpha^{h}\cdot\beta)$ . A product of two (pairs of) connections $\alpha$ and $\beta$ in the above sense willbe simply denoted by $\alpha\cdot\beta$ (or $\alpha\beta$ ) when it does not cause any confusion.

$i^{\mathrm{F}\mathrm{r}}\mathrm{o}\mathrm{m}$ the above definition we are now ready to deal with a system of K-K bi-unitaryconnections which is closed under direct sum, product, conjugation and irreducible de-composition. The precise definition is given ill the following.

118

$\mathcal{G}$

$\mathcal{G}$ $\mathcal{H}$

$\mathcal{H}$

Figure 29:

Definition 3.11 Let $K$ be a connected finite bipartite graph and $K\dot{\mathcal{W}}_{K}$ be a set of equiv-alence classes of horizontally conjugate pairs of K-K $\mathrm{b}\mathrm{i}$-unitary connections with respectto the equivalence relation ‘vertical gauge choice’. Then a set $K\mathcal{W}_{K}$ is called a system ofK-K $\mathrm{b}i$-unitary connections or $\mathrm{s}\mathrm{i}_{1}\mathrm{n}\mathrm{p}\mathrm{l}\mathrm{y}$ a $I_{1}^{\nearrow}- K\mathrm{b}i$-unitary connection system if it is closedunder direct sum, product, conjugation and irreducible decomposition. Another word‘ fusion rule algebra of K-K $bi$-unitary connections’ will often be used in the same mean-ing. A K-K $\mathrm{b}\mathrm{i}$-unitary connection system is said to be Finite if it contains only finitelymany irreducible connections. The rule of irreducible decomposition of products of twoirreducible connections in a system is called the fusion rule of the system.

Remark 3.12 Note that all the above operations are $u$) $ell$-defined on a set of equivalenceclasses of horizontally conjugate pairs of

$\cdot$

K-K $bi$ -unita $7^{\cdot}y$ connections with respect to theequivalence relation $\ell_{Ve\mathrm{r}}$tical gauge choice’. It should be remarked that if we adopt anequivalence relation ‘total gauge choice’ instead of ‘vertical gauge choice’, composition oftwo connection will not necessarily be well-defined.

Example 3.13 Obviously there are two trivial exalllples of K-K $\mathrm{b}\mathrm{i}$-unitary connectionsystems. One is a system which consists of only trivial (identity) connection as its irre-ducible object. The other is a system consisting of all K-K $\mathrm{b}\mathrm{i}$-unitary connections. Forany given family of K-K $\mathrm{b}\mathrm{i}$-unitary $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}_{0}11\mathrm{S}\mathrm{t}\mathrm{h}\mathrm{e}1^{\backslash }\mathrm{e}$ exists a system generated by them.So the word generator of a $\mathrm{s}\mathrm{y}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{n}\overline{1}$akes sellse in this setting.

Example 3.14 Let $N\subset M$ be a subfactor with finite index and finite depth. Then weobtain a (fiat) $\mathrm{b}\mathrm{i}$-unitary connection $W$ on the four graphs as in Figure 29 by Ocneanu’sGalois functor. Where $\mathcal{G}$ is the principal graph alld $H$ is the dual principal graph. Asusual we denote the (vertical) conjugate connection of $W$ by $\overline{W}$ . The composition oftwo connections $W$ and $\overline{W}=W^{v}$ produce a $\mathcal{G}- \mathcal{G}\mathrm{b}\mathrm{i}$ -unitary connection $W\overline{W}$ and a $\mathcal{H}- \mathcal{H}$

$\mathrm{b}\mathrm{i}$-unitary connection $\overline{W}W$ . A systel]] of $\mathcal{G}- \mathcal{G}\mathrm{b}\mathrm{i}$-unitary connections generated by $W\overline{W}$

is finite by the finite depth assumption. This can be regarded the same system as N-$N$ bimodules arising from the subfactor $N\subset M$ . Another system of $\mathcal{H}- \mathcal{H}$ connectionsgenerated by a connection $\overline{W}W$ also corresponds to a system of M-M bimodule arisingfrom the subfactor. This shows that for every subfactor with finite index and finite depthwe can associate two systems of $\mathrm{b}\mathrm{i}$-unitary connections. We will obtain many non-trivialfinite systems of connections in this way.

Example 3.15 Another fundamental example is a system generated by one bi-unitaryconnection, i.e., a singly generated system. Let $(I\iota^{r}, *_{K})$ and $(L, *_{L})$ be two connectedfinite bipartite graphs with the same Perron-Frobenius eigenvalue. Any K-L bi-unitaryconnection $W$ yields a hyperfinite $\mathrm{I}\mathrm{I}_{1}$ subfactor $N\subset M$ by a string algebra construction.This connection $W$ generates a systenl of four kinds of generalized open string $\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{d}\mathrm{U}\mathrm{l}\mathrm{e}\mathrm{s}$

119

([1]) which has the same fusion rule as the $\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}}\mathrm{e}\ln$ of $\mathrm{b}\mathrm{i}\mathrm{n}1\mathrm{o}\mathrm{d}_{\mathrm{U}\mathrm{l}\mathrm{e}}$ arising from the subfactor$N\subset M$ . By looking at the corresponding $\mathrm{b}\mathrm{i}$-unitary connections, we get a system of fourkinds of connections, i.e., K-K, K-L, L-K and L-L connections. Especially we obtainK-K and L-L connection system in this way. If the subfactor $N\subset M$ has finite depth,then both systems will be finite. And if it has infinite depth, they become infinite systems.

Remark 3.16 The procedure in Example 3.15 looks similar to that of Example 3.14. Ac-tually the former is the special case of the latter. Here we remark that the latter is muchmore general because the connections appear in the former example is always flat andinfinite system can not be obtained by the former procedure. Moreover a fiat connectionobtained by the Galois functor is very special because they have (dual) principal graphs asthe four graphs. The following example $show\mathit{8}$ this speciality of fiat connections obtainedby the Galois functor. Flat $connecti_{on}\mathit{8}$ obtained by Sato’s procedure ([29, Theorem 2.1])which is a generalization of the example of [8] show that any finite depth subfactor gener-ated by a (not necessarily flat) $bi$-unitary connection $W$ can be reconstructed by a differentflat connection $W_{f}\cdot W$ which is a horizontally composed connection by $it\mathit{8}$ flat part con-nection $W_{f}$ (See [29], [30]). Hence there are many examples of finite depth $\mathit{8}ubfact_{\mathit{0}}rs$

which are constructed by a flat connection that does not come from the Galois functor.Example 3.17 Let $K$ be one of the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ . It is known thatthere are at most two non-equivalent $\mathrm{b}\mathrm{i}$-unitary connections on the four graphs whichare all the same graph $K$ and trivially connected. (There are only one non-equivalentconnections in the case of $\mathrm{A}_{n}$ and exactly two mutually colnplex conjugate non-equivalentconnections in the case of $D_{n},$ $E_{6,7,8}.$ ) We call $\mathrm{t}\mathrm{h}\mathrm{e}\ln$ fundamental $co\mathrm{n}$nections of the A-D-$E$ Dynkin diagrams. We denote one of them by $W$ and the other by $\overline{W}$ . Let $N\subset M$ bea subfactor constructed by $W$ in the horizontal direction.

First we consider the system generated by a single connection $W$ . In this case thesystem is finite and has the same fusion rule as that of the system of bimodules arisingfrom the subfactor $N\subset M$ .

Next consider the system generated by the two $\mathrm{b}\mathrm{i}$-unitary connections $W_{\Gamma}$ and $\overline{W}$ . Inthis paper we will mainly deal with this systen). It turns out that this system is finite andall the irreducible K-K $\mathrm{b}\mathrm{i}$-unitary connections appear in this system. (See section 5.)

Remark 3.18 It $i_{\mathit{8}}$ a remarkable fact that there are only finitely many K-K irreducible$bi$-unitary connections on the Dynkin diagrams $I\mathrm{i}^{r}$ . This is one of the very special proper-ties of the Dynkin diagrams. We should compare it to the following example. Consider thecase when all the four graphs are the same graph $a\mathit{8}$ the principal graph of Goodman-de laHarpe-Jones subfactor with index $\mathit{3}+\sqrt{3}$ (see Figure 30) which arise from an embeddingof the $A_{11}\mathit{8}tring$ algebra to that of $E_{6}([\mathit{1}\mathit{4}], [\mathit{2}\mathit{5}])$ . In this case there exists one param-eter family of (hence uncountably many) non-equivalent $bi$-unitary $connecti_{on}\mathit{8}$ on thefour $graph_{\mathit{8}}$ . These connections are automatically irreducible by the criterion of Asaeda-Haagerup [1, Corollary 2, Section 3] This means that even if we fix not only the twohorizontal graphs but all the four graphs it can happen that uncountably many irreduciblenon-equivalent connections exist on the $graph_{\mathit{8}}$ . The author would like to $expres\mathit{8}$ his thanksto Y. Kawahigashi for pointing out this example.

Remark 3.19 In this paper we mainly deal with the case when the graph $I\backslash ^{r}$ is one ofthe A-D-E Dynkin diagrams. It is natural to take the $usua,ldi\mathit{8}tinguiShedverteX*_{K}(i.e.$ ,

120

Figure 30: The principal graph of the $\mathrm{G}_{\mathrm{o}\mathrm{O}}\mathrm{d}\mathrm{n}1\mathrm{a}\mathrm{n}$ -de la Harpe-Jones subfactor

the vertex with smallest Perron-Frobenius eigenualue) in these cases and we do so in thefollowing. We denote a hyperfinite $II_{1}$ factor generated by the string algebra on the graph$K$ with the starting point $*_{K}$ by the same notation $I\iota’$ . For a reversed graph $\overline{I\mathrm{c}’}$ we willtake the vertex next to $*_{K}$ as a starting point of string algebra and again we denote itsgenerating factor by the same $\overline{l\mathrm{i}^{\Gamma}}$ . Then by generalized open string algebra construction$([\mathit{1}f)$ we can associate two different bimodules for a given horizontally conjugate pairof K-K $bi$-unitary connections. Again we deal with the pair of generalized open stringbimodules as a corresponding bimodule to the onginal pair of connections. By working onthis correspondence between pairs of connections and pairs of bimodule, we can show thatFrobenius reciprocity holds for the system of connections as in the next proposition.

Proposition 3.20 Let $K,$ $L$ and $\mathit{1}\mathcal{V}I$ be three connected finite bipartite graphs with thesame Perron-Frobenius eigenvalue. Let $K\alpha L,$ $L\beta_{M}$ and $\kappa\gamma_{\Lambda^{\prime I}}$ be three (pairs of) irreducible$bi$-unitary connections which are K-L, L-M and K-M respectively. If $\gamma$ appears $n$ timesin the composite connection $\alpha\beta$ , then $\alpha appear\mathit{8}n$ times in $\gamma\overline{\beta}$ and $\beta$ appears $n$ times in

$\overline{\alpha}\gamma$ .

Proof Choose and fix distinguished vertices from even and odd vertices of each graphs$K,$ $L$ and $M$ . Then apply the correspondence between (pairs of) $\mathrm{b}\mathrm{i}$-unitary connectionsand (pairs of) generalized open string bimodules. We get the result from the

$\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{s}\square$

reciprocity for bimodules ([22], [13, Section 9.8]).

Let $\alpha$ be a K-L $\mathrm{b}\mathrm{i}$-unitary connection. Then from the rule of irreducible $\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\ln_{\mathrm{P}^{\mathrm{O}}}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$

of finite product connections $Kid_{K}\cdot K\alpha L^{\cdot}L\overline{\alpha}K\ldots\kappa\alpha_{L}$ (or $L\alpha_{K}$ ) we get a graph whichis similar to the principal graph of a subfactor. Here $I\mathrm{c}^{\prime id}IC$ denotes an identity K-Kconnection. We call this graph the principal fusion $grapl_{l}$ of a connection $\alpha$ . Then wecan show the following by Asaeda-Haagerup’s criterion of irreducible decomposition ofconnections [1, Claim 1, Section 3]. This is wluat we observed in Example 3.14 and 3.15.

Proposition 3.21 Let $\alpha$ be a K-L $bi$ -unitary connection. Then the principal $fu\mathit{8}i_{on}$ graphof $\alpha$ is the same as the principal graph of the subfactor constructed from the connection

$\alpha$ .

4 Correspondence between connections and $*$-representations of double tri-angle algebras

Let a graph $K$ be one of the Dynkin diagralns $A_{n},$ $D_{n},$ $E_{6},\tau,8$ and $(A, *)$ be the doubletriangle algebra corresponding to the graph $I\iota’$ with the convolution product. This is afinite dimensional $C^{*}$ -algebra and is isomorphic to a finite direct sum of matrix algebrasas follows.

$(A, *)\cong\oplus_{i\in I}H_{i}\otimes\overline{H_{i}}\cong\oplus_{i\in I}\mathrm{E}\mathrm{n}\mathrm{d}(H_{i})$ .

121

$=$$\sum_{i}$

(coef) (i)

Figure 31:

Here the index $i’ \mathrm{s}$ are labelled by nlinilnal central projections in $(A, *)$ and $H_{i}’ \mathrm{s}$ arecorresponding finite dimensional Hilbert spaces. This identification of elements in $(A, *)$

is written as an extension of recoupling as in Figure 31.Now we give the following theorem which lneans that we have only to find all the

minimal central projections in $(A, *)$ in order to classify all irreducible K-K bi-unitaryconnections.

Theorem 4.1 There is $one-t_{\mathit{0}}$ -one correspondence between unitary equivalence classesof irreducible $matri_{C}ial*$ -representations of the K-K double triangle algebra $(A, *)$ andequivalence $cla\mathit{8}seS$ of irreducible K-K $bi$-unztary connections.

Remark 4.2 Equivalence relation on $bi$-unitary $connecti_{on}\mathit{8}$ considered in this theorem$is$ ‘vertical gauge choice’. But it coincides $u$)$ith$ ttotal gauge choice’ in this case as westated in Remark 3.5. (See Lemma 3.6.)

Proof A proof for the construction of a $\mathrm{b}\mathrm{i}$-unitary connection for a $\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}*$ -representationand vice versa is shown in [24, Section 15]. So we only have to show that two unitarilyequivalent $*$-representation give two equivalent $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{S}$ up to vertical gauge choiceand vice versa. It is easy to see that two equivalent connections up to vertical gauge choicegive two unitarily equivalent $*$-representation because the unitary matrix which transferone representation to the other is given by the unitary $11\overline{1}$atrix of gauge choice. For theother direction, first we have to find the vertical edges connecting two horizontal graphs $K$ .Let $\Phi$ be a $\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{C}\mathrm{i}\mathrm{a}\mathrm{l}*$ -representation of the K-K double triangle algebra $(A, *)$ . Becauseelements $a_{x,y}\in A(x, y\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}K)$ as in Figure 32 are $\mathrm{n}\overline{1}$utually orthogonal projections, thematrices $\Phi(a_{x,y})$ are diagonalized with only $0$ and 1 in the diagonal entries by a certainunitary. We draw edges connecting the vertices.$r$ and $y$ with the same nulnbers as thatof 1 in the diagonalized matrix $\Phi(c\iota_{x,y})$ . In this way we get the vertical edges conllectingthe two horizontal graphs. We label these vertical edges by sonle index set A. Then wedefine a connection value of a rectangle as in Figure 33 by the number $\Phi_{\lambda,\mu}(\xi\Theta^{\mathrm{x}}\eta)$ for$\xi,$ $\eta\in \mathrm{E}\mathrm{s}\mathrm{s}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}^{()}n(K)$ and $\lambda,$ $\mu\in$ A. Here $\xi\otimes\eta$ is an $\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$ in $A$ as in Figure 34 and$\Phi_{\lambda,\mu}(\xi\otimes\eta)$ represents a $(\lambda, \mu)$ -th entry of the $\mathrm{n}1\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}_{\mathrm{X}}\Phi(\xi\otimes\eta)$ . If we restrict the mapfrom $A$ to the complex numbers $\mathrm{C}$ defined as above to $\mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}\mathrm{t}}1_{1(K}^{()}1$ ) $\otimes \mathrm{E}\mathrm{s}\mathrm{s}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}^{(1)}(K)$ ,we get the connection map. Now the $\mathrm{b}\mathrm{i}$-unitarity condition of this connection followseasily from the fact that $\Phi$ is a $*$ -representation. (See [24, Section 15].) Note thatin this procedure to get a connection $W^{\Phi}$ fronu a given $*$-representation $\Phi$ , it is easyto see that if the $\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\dot{\mathrm{t}}\mathrm{i}_{\mathrm{o}\mathrm{n}}\Phi \mathrm{i}’ \mathrm{s}$ reducible, then the connecting vertical edges aswell as the connections $W^{\Phi}$ defined on the four $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}\mathrm{s}\backslash$ decompose into some irreduciblecomponents which corresponds to the irreducible components of the $*$-representation $\Phi$ .So the irreducibility is preserved by this correspondence. Now it is easy to see that the

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$\mathrm{m}$

$a_{x,y}$$\equiv$

Figure 32:

$\xi$

$u_{\lambda,\mu}(\xi, \eta)=$ $\lambda a\downarrow c$ $db\downarrow\mu$

$|\overline{/}$

Figure 33:

two unitarily $\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}*$ -representation gives the two equivalent $\mathrm{b}\mathrm{i}$-unitary connectionsup to vertical gauge choice. Moreover one will notice $\mathrm{t}\mathrm{l}\mathrm{T}\mathrm{a}\mathrm{t}$ a $\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}*$-representation

$\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}_{\square }\mathrm{S}$

rise to two horizontally conjugate connections at the salne time.

Remark 4.3 The above correspondence in Theorem 4.1 hold true for the case of K-Ldouble triangle algebras and K-L $bi$-unitary connections. The proof is exactly the same asthe proof of Theorem 4.1. But we remark that we do not haue minimal central $pro_{J^{ect}}ions$

$p_{k}^{\pm}$ as in Section 2 in the case of K-L double triangle algebras when $K\neq L$ .

Applying the above theorem to the concrete $*$-representation corresponding to theminimal central projection $p_{1}^{\pm}$ as in Section 2, we get the following important result.

Corollary 4.4 Let $K$ be one of the Dynkin $diagram\mathit{8}A_{n},$ $D_{n},$ $E6,7,8$ . The minimal centralprojections $p_{1}^{\pm}$ of the K-K double triangle algebra correspond to the two mutually complexconjugate (flat) $bi$-unitary connections on the four graphs which are all the same Dynkindiagram $Ka\mathit{8}$ in 35. In particular $p_{1}^{+}=p_{1}^{-}$ in the case of the Dynkin diagram $\mathrm{A}_{n}$ .

Proof By looking at the shape of the nlinimal central projections $p_{1}^{\pm}$ , the vertical graphsof the corresponding irreducible K-K connections are the graph $K$ itself. Because we knowthat there are two mutually complex conjugate non-equivalent connections on the fourgraphs as in Figure 35 when the graph $I\iota’$ is one of $D_{n},$ $E_{6,7,8}$ . And there is only one$\mathrm{b}\mathrm{i}$-unitary connection when the graph $K$ is one of $A_{n}$ . (See [13, Theorem 11.22]. Weremark that Theorem 11.22 in [13] states the isomorphic classes of connections whereas itis also true for the equivalence classes.) The difference of two connections corresponding

$\xi\otimes\eta$ $\equiv$

Figure 34:

123

$I\iota’$

$K$ $K$

$I\iota’$

Figure 35:

Figure 36:

to $p_{1}^{+}$ and $p_{1}^{-}$ is shown by the relation in Temperley-Lieb recoupling theory as in Figure36 which represents the difference of positive and negative crossing. Where $\epsilon$ is a complexnumber given by $ie^{i\pi/}2h$ with the Coxeter number $h$ . In particular it shows that these

$\mathrm{t}\mathrm{w}\mathrm{o}\square$

connections are mutually complex conjugate.

$i^{\mathrm{F}\mathrm{r}}\mathrm{o}\mathrm{m}$ Theorem 4.1 every $\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$ central projection $p$ in the K-K double trianglealgebra corresponds to an irreducible K-K connection $W_{p}$ . By the definition of productof two connections and the above correspondence, the. product of two minimal centralprojections $p$ and $q$ corresponds to the product of two irreducible $\mathrm{b}\mathrm{i}$-unitary connections$W_{p}$ and $W_{q}$ . So by decomposing the product connection $W_{p}\cdot W_{q}$ into irreducible ones andusing the correspondence between. irreducible connections and minimal central proj ections,we get a linear combination of minimal central projections with positive integer coefficient.This means that the center of the K-K double triangle algebra $Z=Z(A, *)$ is closedunder the . product operation. And this shows the fact that the fusion rule of K-K bi-unitary connections is given by the . product of corresponding minimal central projections.So we get the following.

Corollary 4.5 Let $I\mathrm{i}’$ be one of the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ . Then the fusion rulealgebra of K-K $bi$-unitary connections is isomorphic to the center 2 of the K-K doubletriangle algebra $(A, *)$ with . product, $i.e$ . $(Z, \cdot)$ .

5 Classification of irreducible $\mathrm{b}\mathrm{i}$-unitary connections on the Dynkin diagrams

5.1 Classification of irreducible A-A $bi$-unitary connection$1S$ .

Let $A$ be one of the Dynkin diagran] $A_{n}$ . We first classify all (irreducible) A-A bi-unitary connections.

Proposition 5.1 Let $\mathcal{G}0,$ $\mathcal{G}_{1},$ $\mathcal{G}2,$ $\mathcal{G}_{3}$ be the four graphs connected $a\mathit{8}$ in Figure 22. Supposethat both the upper graph $\mathcal{G}_{0}$ and the lower graph $\mathcal{G}_{2}$ are $A$ and suppose there $\iota s$ a bi-unitary connection on the four graphs. Then the connecting vertical graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{3}$ are

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uniquely determined by the $ini$tial condition, $i.e.$ , the condition of edges connected to thedistinguished $vertex*of$ the upper graph A. Moreover such a connection is unique up tovertical gauge choice.

Proof Because the string algebra on the graph $A$ is generated only by Jones projections,the vertical graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{3}$ are uniquely determined by looking at the dimension ofessential paths with starting point $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{0}\mathrm{n}\mathrm{d}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$ to the initial condition, i.e., the verticesof the lower graph $A$ connected to the distinguished vertex of the upper graph A. (See[13, Section 11.6].) The connection on the four graphs can be decomposed into irreducibleones. And an irreducible A-A connection have only one initial edge by the criterion in[1, Section 3, Claim 1] and the fact tluat it is autolnatically flat ([13, page 593]). Thechoice of the initial edge is one-to-one correspondent to the vertex of the (lower) graph $A$ .Hence the uniqueness (up to vertical gauge choic.e) of the connections on the four graphsis proved by the uniqueness of irreducible connections corresponding to each vertex of$A$ . We know that there is at least one $\mathrm{b}\mathrm{i}$-unitary connection on the four graphs with aninitial edge corresponding to each vertex of the lower graph A. (Note that a correspondingcommuting square is given in [25].) So the proof will end to show that these are the onlyirreducible $\mathrm{b}\mathrm{i}$-unitary connections up to vertical gauge choice. This is done by using thecorrespondence between $*$-representations of the A-A double triangle algebras and A-A$\mathrm{b}\mathrm{i}$-unitary connections. More precisely we llave to sllow the following equality.

$a \in\sum_{\mathrm{e}\mathrm{r}\mathrm{t}A}(_{k0}^{n-1}\sum_{=}\dim \mathrm{E}_{\mathrm{S}}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}(oAk\backslash ))2=\sum_{=0}n-1k(_{a\in \mathrm{v}_{\mathrm{e}}\mathrm{I}\mathrm{t}A}\sum \mathrm{d}\mathrm{i}\ln \mathrm{E}\mathrm{S}\mathrm{S}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}_{a}(k)A)2$

This equality can be easily obtained by a direct computation. $\square$

Remark 5.2 The above method also works for the classification of irreducible A-K bi-unitary connections for arbitrary Dynkin diagrams $I\iota’$ with the same Coxeter number asA. The only different point is the last equaldy concerning the dimensions of the doubletriangle algebras. In the case of general Dynkin diagram $I\iota’$ , we have to show the followingequality instead of the above one.

$x \in \mathrm{v}\sum_{\mathrm{e}\mathrm{r}\mathrm{t}K}(_{k=}\sum^{m}\dim \mathrm{E}\mathrm{s}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}^{(}k0x)K\mathrm{I}^{2}=\sum_{k=0}^{n1}(_{a\in \mathrm{V}\mathrm{e}}\sum_{\mathrm{t}\mathrm{r}A}$dinl $\mathrm{E}\mathrm{s}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}_{a)}(k.)A(_{x\in \mathrm{e}\mathrm{r}}\sum_{\mathrm{V}\mathrm{t}K}\dim \mathrm{E}\mathrm{S}\mathrm{S}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}_{x}^{(k})K)$

Here $m$ is the maximal length of essential paths on $A$ and $IC$ which is the same as (theCoxeter $number$) $-2$ . This also can be shown by a direct computation in each case. Buthere we give another proof based on estimates of the global index in the following.

Remark 5.3 The uniqueness of irreducible A-K $bi$ -unitary connection corresponding toeach vertex of $K$ does not seem to be obvious though we know the uniqueness of corre-sponding commuting square up to isomorphism. Here we remark that an isomorphic class

of commuting $\mathit{8}quarecorre\mathit{8}ponds$ an uomorphic class of connections (See [13, Definition10.11] for the definition of isomorphic connections) and it does not imply the uniqueness

of equivalent connections up to (vertical) gauge choice.

5.2 Classification of irreducible A-K $bi$-unitary $com$lections.

Let $A$ be one of the Dynkin diagranls $An$ and $I\iota’$ one of the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$

with the same Coxeter number as $A$ . Before going into the details of the classification,

125

we will show that a simple consideration on the $\mathrm{f}\mathrm{u}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}$ rule algebra leads an $\mathrm{i}_{\ln_{\mathrm{P}^{\mathrm{o}\mathrm{r}}}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$

consequence, that is, the system of $\mathrm{b}\mathrm{i}$-unitary connectiolls which consist of all irreducibleA-A, A-K, K-A and K-K connections are finite. It llleans that the numbers of all equiv-alence classes of irreducible A-K and K-K $\mathrm{b}\mathrm{i}$-unitary connections are finite. Moreover wecan measure its size explicitly. To sllow this first we need the next lemma.

Lemma 5.4 Let $A$ and $I\mathrm{i}^{r}$ be as above. Then all irreducible K-K (resp. A-A) bi-unitaryconnections are obtained from the product $K\overline{\alpha}A$ $A\beta\kappa$ (resp. $A\alpha_{K}\cdot K\overline{\beta}_{A}$ ) for $\mathit{8}ome$ twoirreducible A-K $bi$-unitary connections $\alpha$ and $\beta$ .

Proof We give a proof for the case of K-K $\mathrm{b}\mathrm{i}$-unitary connections because the sameproof also works for the case of A-A $\mathrm{b}\mathrm{i}$-unitary connections. This is an easy consequenceof Frobenius reciprocity. Let $K^{W}K$ be any irreducible K-K $\mathrm{b}\mathrm{i}$-unitary connection. Thentake (any) irreducible A-K $\mathrm{b}\mathrm{i}$-unitary connection $A\alpha_{K}$ and make a product of them.Take an irreducible A-K connection $A\beta_{K}$ in the irreducible decomposition of the product$A\alpha_{KK}.w_{K}$ , i.e., we have $A\alpha w_{K}\succ A\beta\kappa$ . So we get $I_{1}^{\prime\overline{\alpha}}\beta_{K}\succ K^{W}K$ by Frobenius

$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{P}^{\Gamma \mathrm{O}\mathrm{C}}\mathrm{i}\mathrm{t}\mathrm{y}\square$

.

Theorem 5.5 Let $l\mathrm{i}’$ be one of the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ . Then the numbersof all equivalence classes of irreducible A-K and K-K $bi$-unitary connections are finite.Moreover, they have the same global index $a\mathit{8}$ that of the system of all irreducible A-A$bi$-unitary connections.

Proof The case when the graph $I\mathrm{c}’$ is $A_{n}$ is shown in the previous section. So we considerthe other cases. The system of all A-A $\mathrm{b}\mathrm{i}$-unitary connections are obtained from a finiteset of irreducible A-K connections by the previous lemnla. We choose and fix such a finiteset and consider the system generated by one A-K $\mathrm{b}\mathrm{i}$-unitary connection $A^{W}K$ which isthe (finite) direct sum of all the A-K connections we have chosen. It was shown thatthe set of all irreducible A-A connections are finite. So this $\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}}\mathrm{e}\ln$ contains only finitelymany different irreducible A-K connections because of the local finiteness of the principalfusion graph of the generator $A^{W}K$ .

We claim that all irreducible A-K connections appear in this system. Otherwise wehave an $\mathrm{i}\mathrm{r}$.reducible A-K connection $A^{\tilde{L}}.K$ which do not appear in this system. If we takean A-K connection $w\oplus z$ as an generator, we get a different system having the same setof irreducible A-A connections and strictly larger set of irreducible A-K connections thanbefore. Thus we get two systems of four kinds of connections consisting of irreducible A-A,A-K, K-A and K-K connections which are both generated by one A-K connection. Nowit is easy to see that proof of the equality $\sum_{N}\lambda_{N}[_{N}X_{N}]=\sum_{N}Y_{M}[_{N}Y_{M}]=\sum_{M}Z_{\Lambda i}[_{M}Z_{M}]$

for the estimates of global index for subfactor $N\subset M$ still works in the case of singlygenerated connection system. Here the sulnnlations run over all irreducible $\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{d}_{\mathrm{U}\mathrm{l}\mathrm{e}}$

appear in the system generated by $N^{\mathrm{j}1I}\Lambda,I$ . (See [13, Proposition 12.25] for the proof ofsubfactor case.) So we get a contradiction from the estilnates of the global index of thetwo systems. Hence the system must contain all the irreducible A-K connections.

Now the same argument shows that we have finitely many irreducible K-K connec-tions and these are the all irreducible K-K connections by the previous lemma. So thesystem of four kinds of connections consisting of all irreducible A-A, A-K, K-A and K-Kconnections is generated by one A-K connection $w$ . Applying the estimates of the

$\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}1\coprod$

indices of this system we get the result.

126

By this estimates of global index we can easily classify all irreducible A-K bi-unitary

connec..tions as $\mathrm{i},\mathrm{n}$ the following proposition.. : : ... . $r:$

...

Proposition 5.6 Let $\mathcal{G}0,$ $\mathcal{G}_{1},$ $\mathcal{G}2,$ $\mathcal{G}_{3}$ be the four graphs connected.. as in Figure 22. Suppose

that the upper graph $\mathcal{G}_{0}$ is $A$ and the lower graph $\mathcal{G}_{2}$ is $K$ and $\mathit{8}uppose$ there is a bi-

unitary connection on the four graphs. Then the connecting vertical graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{3}$ areuniquely determined by the initial condition, $i.e.$ , the condition of edges connected to thedistinguished vertex of

$t.heg.ra.ph\backslash .\cdot.,’\backslash$

A.$M.oreov\wedge\cdot,.\cdot.$

‘

er$suc_{\mathfrak{i}}.h..a.:conn.e,c.:t‘.i\mathit{0}n_{l},$$?s..uni..q_{\mathrm{f}}’\backslash _{\iota\backslash }\cdot!\wedge\cdot,\backslash \cdot.u.ej.\cdot$

.

up$to.v;$

.ertical

gauge choice. .‘

Proof The proof of the first assertion is exactly the same as Proposition 5.1. So wehave only to show the uniqueness of irreducible connections corresponding to each vertexof $K$ . Again we know that there is at least one $\mathrm{b}\mathrm{i}$-unitary connection on the four graphs

with an initial edge corresponding to each vertex of $I\mathrm{c}’([25])$ . So we show that these arethe only irreducible $\mathrm{b}\mathrm{i}$-unitary connections up to vertical gauge choice.

In the case of $K=D_{n}$ , we know that there is an irreducible A-K connections wit.h.index $\sqrt{2}$ which correspond $\backslash \mathrm{t}\mathrm{o}$ the $\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}*\mathrm{o}\mathrm{f}D_{n}$. We denote it by $A\alpha_{K}$ . Take one of the

two non-equivalent fundamental connections on $D_{n}$ and denote it by $K^{W}K$ . Take a finiteproduct $\alpha w\overline{w}\cdots w$ (or $\overline{w}$ ) and decompose them. In this way we get some irreducible A-K

connections. We remark that the fusion graph with initial vertex $\alpha$ and generator $w$ has

the same Perron-Frobenius eigenvalue as the index of the connection $w$ . So it must be oneof A-D-E Dynkin diagrams. It is easy to see that irreducible connections corresponding

to any choice of the initial vertex of $I\backslash ^{r}$ appears in this procedure. So the graph has

vertices at least as many as that of $D_{\eta}$ . $i$From the estimates of global index, these areall the irreducible A-K connections because of the equality $|A_{2n+1}|=2\cdot|D_{n}|$ . Here $|I\mathrm{i}’|$

represents the global index $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{0}\mathrm{n}\mathrm{d}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$ to all the vertices of the graph $l\mathrm{i}’$ .In the case of $K=E_{6,7,8}$ , the salne proof as in the case of $D_{n}$ works. So we have only to

show the following estimates of global $\mathrm{i}_{\mathrm{I})}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{S}$, i.e. $|E_{6}|=|\alpha_{E_{6}}|^{2}\cdot|A_{11}|,$ $|E_{7}|=|\alpha_{E_{7}}|^{2}\cdot|A_{17}|$ ,

and $|E_{8}|=|\alpha_{E_{8}}|^{2}\cdot|A_{29}|$ . Here $\alpha_{I\mathrm{c}’}$ represents a connection corresponding to an initial

edge connected to the distinguished $\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}*\mathrm{o}\mathrm{f}$ the graph $I_{1}’$ and $|\alpha_{K}|$ denotes its index.

These are shown by Wenzl’s index formula [14, Theorem 4.3.3]. Actually the square of

the indices of connections $\alpha_{K}$ as above are the same as the indices of the corresponding

GHJ subfactors, which are exactly the quotient of two global indices of $I\mathrm{t}^{\mathcal{F}}$ and $A$ , i.e.$|\alpha_{K}|^{2}=|K|/|A|$ for $K=E_{6,7,8}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{l}$ Wenzl’s index forlllula. So the above equalities

$\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\square$

.

$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$ Proposition 5.1 and Proposition 5.6 we get the following theorem.

Theorem 5.7 Let $K$ be one of the Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ . There $i\mathit{8}$

$a$ one.-. to-onecorrespondence between vertices of the graph $I\mathrm{c}^{-}$ and equivalence classes of irreducible A-K$bi$-unitary connections.

5.3 Classification of irreducible K-K $bi$-unitary connections.

$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$ the one-to-one correspondence between $\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$ central projections of the K-$K$ double triangle $\mathrm{a}\mathrm{i}_{\mathrm{g}\mathrm{e}}\mathrm{b}\mathrm{r}\mathrm{a}$ and K-K $\mathrm{b}\mathrm{i}$-unitary connections, for every minimal centralprojection $p$ of the K-K double triangle algebra we can associate an index of the subfactorgenerated by the corresponding connections. We call the square root of the index of the

subfactor corresponding to a $\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{l}$ central projection $p$ an index of the projection $p$

and we denote it by $\mathrm{d}(p)$ . Because two equivalent $\mathrm{b}\mathrm{i}$-unitary connections give rise to

127

an isomorphic subfactor, this definition is well-defined. Moreover if two $\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$ centralprojections $p$ and $q$ coincide, they $11\mathrm{l}\mathrm{U}\mathrm{s}\mathrm{t}$ have the sanue index, i.e. $\mathrm{d}(p)=\mathrm{d}(q)$ .

For the case of the Dynkin diagrallls $K$ we have $\mathrm{s}_{1}\supset \mathrm{e}\mathrm{c}\mathrm{i}\mathrm{a}1$ central projections $\Psi_{+}$ and$\Psi$-which is called chiral projectors (see Section 2.3). We call the subset of $(Z, \cdot)$ whichconsists of minimal central projections contained in $\Psi_{+}$ (resp. $\Psi_{-}$ ) chiral left part (resp.chiral right part). Because the chiral left part (resp. chiral right part) coincide with theset of minimal central projections appears in the system generated by $p_{1}^{+}$ (resp. $p_{1}^{-}$ ) theyform two fusion rule subalgebras of $(Z, \cdot)$ . The intersection of chiral left part and chiralright part is called ambichiral part of the fusion rule algebra $(Z, \cdot)$ and it correspondsto the ambichiral projector $\Psi_{\pm}$ . In the following we use the notations $Z_{l},$ $\mathcal{Z}_{r}$ and $Z_{a}$ torepresent the fusion rule subalgebras of chiral left part, chiral right part and anlbichiralpart respectively.

Proposition 5.8 Let $K$ be one of Dynkin diagrams $A_{n},$ $D_{n},$ $E_{6,7,8}$ . Suppose we have afusion rule subalgebra $B$ of the fusion rule algebras of all irreducible K-K connections2. Then 2 $decompoSe\mathit{8}$ into left cosets of $B$ and right cosets of $B,$ $i.e$ . we have subsets$X,$ $\mathrm{Y}\subset Z$ of irreducible connections (representatives of left and right $co\mathit{8}ets$) such that$\mathcal{Z}=\bigcup_{x\in x^{X\cdot B}}=\bigcup_{y\in Y}B\cdot y,$ $x\cdot B\cap x^{J}\cdot g=\emptyset$ if $x\neq x’\in X$ and $B\cdot y\cap B\cdot y’=\emptyset$ if$y\neq y’\in \mathrm{Y}$ .

Proof We will give a proof for the left cosets. We have only to show the following;$x\cdot B\cap x’\cdot B=\emptyset$ for irreducible $x,$ $x’\in Z$ , then $x\cdot B=x’\cdot B$ . Suppose we have$x\cdot B\cap x’\cdot B=\emptyset$ for $x,$ $x’\in$ Z. Then there are irreducible K-K connections $b,$ $b’\in B$

and $z\in Z$ , such that $x\cdot b\succ z$ and $x’\cdot b’\succ z$ . Hence $x\cdot B\supset x\cdot b\cdot B\supset z\cdot B$ and$x’\cdot B\supset x’\cdot b’\cdot B\supset z\cdot B$ holds. $\mathrm{F}\mathrm{r}\mathrm{o}\ln$ the Frobenius reciprocity, we have $z\cdot\overline{b}\succ x$ and$z\cdot\overline{b}’\succ x’$ . So the converse inclusions $z\cdot B\supset z\cdot\overline{b}\cdot B\supset x\cdot B$ and

$z\cdot B\supset z\cdot\overline{b}’\cdot B\supset x’\cdot B\coprod$

holds. Thus we have $x\cdot B=x’\cdot B=z\cdot B$.

Remark 5.9 It is easy to see that this proposition holds true for more general fusion rulealgebras such as those treated in Hiai-Izumi [$\mathit{1}\mathit{5}J$ . We only need the property of Frobeniusreciprocity. For example any fusion rule algebras of bimodule (or $\mathit{8}ectors$) arising fromsubfactors have this coset decomposition property.

The following can be easily shown from Proposition 5.8 and the estimates of globalindices (Theorem 5.5).

Corollary 5.10 If the chiral left part $\mathcal{Z}_{l}$ does not coincide with the chiral right part $Z_{r}$ ,then the principal fusion graphs of minimal central projections $p_{1}^{+}$ and $p_{1}^{-}$ cannot be theDynkin diagram of type A. Conversely if one of the principal fusion $graph_{\mathit{8}}$ of $p_{1}^{+}$ and $p_{1}^{-}$

is the Dynkin diagram of type $A$ , then we have $Z_{l}=Z_{r}=Z_{a}=Z$ .

5.3.1 The case of $A_{n}$

This is done in the previous subsection 5.1. There is $\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}_{\mathrm{o}^{-}\mathrm{o}\mathrm{n}\mathrm{e}}$ correspondence be-tween vertices of the Dynkin diagram $A_{n}$ and irreducible $A_{n^{-}}A_{n}\mathrm{b}\mathrm{i}$-unitary connections.In this case the two minimal central projections $p_{1}^{+}$ and $p_{1}^{-}$ coincide. The fusion rule graphfor the generator $[1]=p_{1}^{+}=p_{1}^{-}$ is given in Figure 40.

5.3.2 The case of $D_{2n+1}$

128

In this case we have gap $(D_{2}n+1)=\infty$ and corresponding recoupling system is $A_{4n-1}$ .So there are two series of mutually orthogonal minimal central projections $\{p_{k}^{+}\}_{k=0},1,2,\ldots,4n-2$

and $\{p_{k}^{-}\}_{k=}0,1,2,\ldots,4n-2$ . Here $p_{0}^{+}$ and $p_{0}^{-}$ coincide and it corresponds to the identity con-nection. Because all $p_{k}^{+}$ arise from $p_{1}^{+}$ by taking . product, this means that the subfactorarising from the connection corresponding to $p_{1}^{+}$ (which is one of the two non-equivalent$\mathrm{b}\mathrm{i}$-unitary connections as in Corollary 4.4) have at least $4n-1$ vertices in its principalgraph. From the index value $\mathrm{d}(p_{1}^{+})$ this is possible only when the principal graph is theDynkin diagram $A_{4n-1}$ . So we luave $Z_{l}=Z_{r}=Z_{a}=\mathcal{Z}$ by Corollary 5.10. Hence theseare all the minimal central projections. $i^{\mathrm{F}}\mathrm{r}\mathrm{o}\ln$ the facts that $p_{1}^{-}\neq p_{1}^{+},$ $p_{1}^{-}\neq p_{4n-3}^{-}$ and$\mathrm{d}(p_{1}^{+})=\mathrm{d}(p_{1}^{-})=\mathrm{d}(p_{4n-3}^{+})=\mathrm{d}(p_{4n}^{-}-3)$ , the lninimal central projection $p_{1}^{-}$ must coincidewith $p_{4n-3}^{+}$ . Hence we get $p_{k}^{-}=p_{4n-}^{+}2-k\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{n}\mathrm{l}$ the fusion rule. The fusion graph of the twogenerator $[1]=p_{1}^{+}$ and $[4n-3]=p_{1}^{-}$ are given as in Figure 41.

Note that we did not use the fact of non-existence of $D_{2n+1}$ subfactors in the aboveargument. So it gives another proof of the non-existence of $D_{2n+1}$ subfactors. It alsoshows that the flat part of $D_{2n+1}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}$ squares are $A_{4n-1}$ .

5.3.3 The case of $D_{2n}$

In this case there are two non-equivalent connections with index 1. One is trivialconnection and the other comes fronl the flip of the two tails of the graph $D_{2n}$ (we denote

it by $\epsilon$ ). We denote the minimal central projections corresponding to the connection $\epsilon$ by$p_{\epsilon}$ .

Because $p_{1}^{+}\neq p_{1}^{-}$ , the fusion graph of chiral left part as well as chiral right part cannot be $A_{4n-3}$ from Corollary 5.10. So they must be $D_{2n}$ except the case $n=5,8$ . Butin the case of $D_{10}$ and $D_{16}$ we have gap $(D_{10})=16$ and $\mathrm{g}\mathrm{a}\mathrm{p}(D_{16}++)=28$ . Hence there is aseries of mutually orthogonal $\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$ central projections $p_{0},p_{1},$ $\ldots,p_{7}+\in Z_{l}$ in the caseof $D_{10}$ and $p_{0}^{+},p^{+}1$ , $\ldots,p^{+}13\in Z_{l}$ in the case of $D_{16}$ . These shows that the fusion graph ofchiral left part (hence chiral right part as well) can not be $E_{6}$ or $E_{8}$ and they lllust be$D_{10}$ and $D_{16}$ themselves.

It is easy to see that $p_{\epsilon}$ does not appear in either $\mathcal{Z}_{l}$ nor $\mathcal{Z}_{r}$ by comparing the indicesexcept the case $D_{4}$ . In the case of $D_{4}$ , the $\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{r}\perp \mathrm{r}\mathrm{u}\mathrm{l}\mathrm{e}$ algebra of even vertices of $D_{4}$ is thecyclic group $\mathrm{Z}_{3}$ . Hence $p_{\epsilon}\not\in Z_{l}\cup Z_{r}$ in this case, either. So we get the coset decolnposition$\mathcal{Z}\supset Z_{l}\cup Z_{l}\cdot p_{\in}$ . But the estimates of the global indices of the both sets $\mathcal{Z}$ and $Z_{l}\cup Z_{l}\cdot p\epsilon$

shows that these are all the irreducible K-K collnectiolls.$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$ the equalities $p_{1}^{-}=p_{\xi j}\cdot p_{1}+=p_{1}^{+}\cdot p_{\mathcal{E}}$ and $p_{\epsilon}^{2}=p_{\epsilon}\cdot p_{\epsilon}=id$ we have $p_{2}^{+}=p_{2}^{-}$ ,

which shows that the even vertices of chiral left and right part coincide from fusion rule of$D_{2n}$ . It is easy to see that the odd vertices of $\mathcal{Z}_{l}$ and $Z_{r}$ does not coincide again from the

fusion rule. So we obtain the fusion rule graph for two generators $[1]=p_{1}^{+}$ and $[1^{\sim}]=p_{1}^{-}$

as in Figure 42.

5.3.4 The case of $E_{6}$

In the case of $E_{6}$ we know that $p_{1}^{+}\neq p_{1}^{-}$ . By $\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{k}\mathrm{i}\mathrm{l}$ at the vertical edges of thecomposite connection corresponding to $p_{1}^{+}$ . $p_{1}^{-}$ , we can see that this connection is irre-ducible from the criterion [1, Corollary 2, Section 3] and the Frobenius reciprocity. This

means that $p_{1}^{-}$ is not in the chiral left part, i.e. $p_{1}^{-}\in Z_{l}$ . So we have coset deconlposition$\mathcal{Z}\supset Z_{l}\cup \mathcal{Z}_{l}\cdot p_{1}^{-}$ (see Proposition 5.8). The (coset) principal fusion graphs for $\mathcal{Z}_{l}$ and$\mathcal{Z}_{l}\cdot p_{1}^{-}$ are one of the Dynkin diagrams $D_{7}$ or $E_{6}$ . We cannot have $A_{11}$ fronl Corollary

5.10. The estimates of global indices show that $|A_{11}|=(3+\sqrt{3})|E6|=(1+|p_{1}^{-}|^{2})|E_{6}|$ .$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$ this together with the following inequality $|E_{6}|<|D_{7}|<|A_{11}|$ the both (coset)principal fusion graphs for $Z_{l}$ and $Z_{l}\cdot p_{1}^{-}$ lllust be $E_{6}$ . This also shows the fiatness of

129

Table 1: Multiplication table for $\cdot\copyright_{N}$ . of the fusion rule algebra of $E_{6}$

$N-N$

$N-M$$\Lambda I-N$

$M-\Lambda I$

Figure 37: The (dual) principal graph $E_{6}$

$E_{6}$ connections (see Proposition 3.21) and that the whole system $\mathcal{Z}$ is generated by thetwo irreducible fundamental connections $p_{1}^{+}$ and $p_{1}^{-}$ . Now it is easy to see the fusion rulegraph for the two generators $[1]=p_{1}^{+}$ and $[1^{-}]=p_{1}^{-}$ is given in Figure 43 from the indexvalues. Here we also give the multiplication table of the $E_{6}$ fusion rule algebra in Table1, where $\rho_{i}’ \mathrm{s}$ correspond to the vertices of $E_{6}$ shown in Figure 37.

5.3.5 The case of $E_{7}$

We have $p_{1}^{+}\neq p_{1}^{-}$ and the irreducibility of the composite connection $p_{1}^{+}\cdot p_{1}^{-}$ can beshown in the same way as $E_{6}$ case. So we have coset decomposition $Z\supset Z_{l}\cup Z_{l}\cdot p_{1}^{-}$ andthe (coset) principal fusion graphs for $\mathcal{Z}_{l}$ and $Z_{l}\cdot p_{1}^{-}$ are one of the Dynkin diagrams $D_{10}$

or $E_{7}$ . (Again we cannot have $A_{18}$ from Corollary 5.10.)In this case the global indices satisfies $|E_{7}|=2\beta(3\beta^{2}-15\beta+18)<|D_{10}|=\beta|E_{7}|<$

$|A_{17}|=2\beta|E_{7}|$ , where $\beta=4\cos^{2}(\pi/18)$ . So we have $|A_{17}|=2\beta|E_{7}|>|G_{1}|+\beta|G_{2}|$ ,where $G_{1}$ and $G_{2}$ denote the (coset) principal fusion $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}1_{1}$ for $Z_{l}$ and $Z_{l}\cdot p_{1}^{-}$ respectively.Hence the equality only happens when $G_{1}=D_{10}$ and $G_{2}=E_{7}$ . This shows that the fiatpart of the $E_{7}$ connections is $D_{10}$ and that the whole system $Z$ is generated by the twofundamental $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}++\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}p_{1}+$ and $p_{1}^{-}$ . Here we claim $p_{2}^{+}\neq p_{2}^{-}$ . If $p_{2}^{+}--p_{2}^{-}$ , from the fusionrule we have $p_{1}\cdot p_{1}\cdot p_{1}^{-}=2p_{1}^{-}+p_{3}^{-}$ . But this is inlpossible because $\dim \mathrm{E}\mathrm{n}\mathrm{d}(p_{1}+$ . $p_{1}^{-)}=$

$\dim \mathrm{H}\mathrm{o}\mathrm{m}(p^{+}1^{\cdot}p_{1}^{+}\cdot p^{-}1 , p_{1}^{-})=2$ by Frobenius reciprocity, which contradicts the irreducibilityof $p_{1}^{+}\cdot p_{1}^{-}$ . Hence we must have $p_{2}^{+}\neq p_{2}^{-}$ . Then by looking at the indices and the fusionrules, we $\mathrm{g}.\mathrm{e}\mathrm{t}$

the fusion graph for the two generators $[1]=p_{1}^{+}$ and (0) $=p_{1}^{-}$ as in Figure43.

5.3.6 The case of $E_{8}$

In this case we also have the coset decomposition $\mathcal{Z}\supset Z_{l}\cup Z_{l}\cdot p_{1}^{-}$ which is shown

130

$(\beta^{2}-2\beta+2)|E_{8}|<|A_{17}|=(2\beta^{2}\neg 4\beta+4)|E_{8}|$, where $\beta=4\cos^{2}(\pi/30)$ .First we show the principal fusion graph of $p_{1}^{+}$ is $E_{8}$ . $i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$ the above coset decompo-

sition, we have the following inequality, $|A_{17}|=(2\beta^{2}-4,\theta+4)|E_{8}|>|G_{1}|+\beta|G_{2}|$ . Here$G_{1}$ and $G_{2}$ denote the (coset) principal fusion graph for $Z_{l}$ and $Z_{l}\cdot p_{1}^{-}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}1}\mathrm{y}$. If$G_{1}=D_{16}$ , there are two possibilities, i.e. $p_{2}^{+}=p_{2}^{-}$ or $p_{2}^{+}\neq p_{2}^{rightarrow}$ . If $p_{2}^{+}=p_{2}^{-}$ holds, tlle evenvertices of the chiral left part $\mathcal{Z}_{l}$ and those of the chiral right part $Z_{r}$ coincide from thefusion rule. And we have the following coset decomposition, $Z\supset Z_{l}\cup Z_{l}\cdot p_{1}^{-}\cup Z_{l}\cdot p_{3}^{-}$

because $p_{3}^{-}\not\in \mathcal{Z}_{l}\cup Z_{l}\cdot p_{1}^{-}$ . But the smallest possible value of the global indices is$|D_{16}|+\beta|E8|+\beta(\beta-2)2|E_{8}|=(\beta^{3}-3\beta 2+3\beta+2)|E_{8}|>|A_{29}|$ and this is impossible. If

$p_{2}^{+}\neq p_{2}^{-}$ holds, then $p_{2}^{-}\not\in \mathcal{Z}_{l}\cup Z_{l}\cdot p_{1}^{-}$ by comparing the indices. So we have the cosetdecomposition $Z\supset Z_{l}\cup z_{lp_{1}^{-}}.\cup \mathcal{Z}_{l}\cdot p_{2}^{-}$ . The smallest possible value of the global indicesis $|D_{16}|+\beta|E8|+(\beta-1)^{2}|E_{8}|=(2\beta 2-3\beta+3)|E_{8}|>|A_{29}|$ . So again this is $\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{p}_{0}\mathrm{s}\mathrm{S}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}$ .Thus the principal fusion graph of $p_{1}^{+}$ must be $E_{8}$ . And this shows the fiatness of $E_{8}$

connections. We label the vertices of the chiral left and right part as in Figure 38.Next we show that the ambichiral part $Z_{a}=Z_{l}\cap Z_{r}$ is $\{\rho_{0}, \rho_{6^{--}}\beta 6\}\sim i^{\mathrm{F}\mathrm{g}\mathrm{a}_{\mathrm{P}(}}\mathrm{r}\mathrm{o}\mathrm{n}1E8)$

$=0-\mathrm{g}\mathrm{a}\mathrm{p}(E8)=10$ , we have two series of mutually orthogonal minimal central projections

$\mathrm{a}\mathrm{r}\mathrm{e}\{p_{k}^{+}\}_{k0,1}\mathrm{t}\mathrm{W}\mathrm{o}\mathrm{p}\mathrm{o}=,2,3,4\mathrm{a}\mathrm{n}_{\mathrm{i}}\mathrm{d}\{pk\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{i}1\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{S},-\}_{k}.=0,1,2,3,4\cdot \mathrm{T}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}_{2}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}1\mathrm{s}\mathrm{o}1\mathrm{a}_{\mathrm{O}}\mathrm{b}\mathrm{e}\mathrm{l}1\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}_{1}p_{k}^{+}=\mathrm{e}.\rho 2=\rho 2\mathrm{o}\mathrm{r}\sim\rho\neq\rho 2\mathrm{f}\mathrm{r}\mathrm{m}\mathrm{t}11\mathrm{e}\mathrm{i}_{1}\mathrm{d}\mathrm{e}\mathrm{x}\mathrm{V}\mathrm{a}1\mathrm{u}\mathrm{e}\mathrm{s}.\mathrm{I}\mathrm{f}\rho_{2}=\rho_{2}-,$

$\mathrm{t}\mathrm{h}\sim \mathrm{d}\beta_{k}\mathrm{a}\mathrm{n}pk\rho_{k}=\mathrm{T}+\sim.\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}$

from the fusion rule (Table 3) we have $\beta 1^{\cdot}\rho 1^{\cdot}\rho 1^{-}=(p_{0}+\rho_{2})\cdot\rho_{1}^{\sim}=2\beta_{1}^{\sim}+\rho 3\sim$. But this isimpossible because $\dim \mathrm{E}\mathrm{n}\mathrm{d}(\rho_{1\rho_{1}^{\sim})}.=\dim \mathrm{H}\mathrm{o}\mathrm{n}\mathrm{u}(\beta_{1}.(\rho_{1}\cdot\rho_{1}^{\sim}), \rho_{1})$ by Frobenius reciprocityand it contradicts the irreducibility of $\rho_{1}\cdot\rho_{1}-$. So $\rho_{2}$ and $\rho_{2}^{\sim}$ do not coincide. Then againfrom the fusion rule we cannot have $\rho_{4}=\rho_{4}^{\sim}$ which contradicts $\rho_{2}\neq\rho_{2}^{\sim}$ . Now if $\rho_{6}\neq\rho_{6^{-}}$

then we can check the following coset $\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{u}\mathrm{p}_{\mathrm{o}\mathrm{s}}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n},$ $\mathcal{Z}\supset Z_{l^{\cup}}\mathcal{Z}l\rho^{-}1\cup \mathcal{Z}_{l}\cdot\rho_{2}^{\sim}\cup \mathcal{Z}_{l}\cdot\rho^{\sim}5$

$\cup Z_{l}\cdot\rho_{6}^{\sim}$ by comparing indices. The slnallest possible value of the global index of thesecosets is $(2\beta^{2}-3\beta+3)|E_{8}|>|A_{29}|$ , which contradicts the estimates of global indices(Theorem 5.5). Hence we must have $\rho_{6}=\rho_{6^{-}}$. It is easy to see that the odd vertices of

$\mathcal{Z}_{l}$ and $Z_{r}$ cannot coincide because of the fusion rule (Table 3) and the fact $\rho_{1}=\rho_{1}^{\sim}$.

Remark 5.11 Here we remark that the multiplication $table\mathit{8}\mathit{1},\mathit{2}$ and 3 are for the system

of bimodules or sectors. But in the case of K-K connection systems we have differentgradings. Actually we have equality $u$) $=\overline{w}f\mathit{0}r$ the two fundamental connections $u$) of A-D-E Dynkin diagrams. This is impossible in the case of bimodules $becau\mathit{8}ew$ correspondsto an $N-M$ bimodule. In the case of sectors self-conjugate sector $\rho_{1}=\overline{\rho}_{1}$ makes sense,but these have different meaning. (; From the fact that the fundamental connection $w$ ,

which $corre\mathit{8}ponds$ to $\rho_{1}$ in the multiplication tables, is self-conjugate in our sense as apair of connections, we can easily get $\rho_{k}$. $=\overline{\rho}_{k}$. for odd $k$ and $\rho_{k}’=\rho_{k}$ for even $k$ in the case

of $E_{6}$ and $E_{8}$ . Again one will notice that $\rho_{k^{\backslash }}’=\rho_{k}$. for even $k$ does not make sense even inthe case of sectors because the left hand side $?S$ in $\mathrm{s}_{\mathrm{e}\mathrm{c}\mathrm{t}}(\Lambda I)$ while the right hand side $i\mathit{8}$ in

$\mathrm{s}_{\mathrm{e}\mathrm{c}\mathrm{t}}(N)$ . But in our setting of K-K connection systems such things can happen. Hencewe can read the multiplication $\overline{\rho}_{1}\cdot\rho_{2}=\overline{\rho}_{1}+\overline{\rho}_{3}$ and $\overline{\rho}_{1}\cdot\rho_{3}=\rho_{2}’+\rho_{4}’$ as $\rho_{1}\cdot\rho_{2}=\rho_{1}+\rho_{3}$

and $\rho_{1}\cdot\rho_{3}=\rho_{2}+\rho_{4}$ respectively for examples in Table 2 and 3.

Now from the fusion rule we know $\rho_{7^{-}}=\rho 6^{\cdot}\rho 1\sim,$ $\rho_{4^{-}}=\rho 6^{\cdot}\rho^{\sim}2$ and $\rho_{3}=\rho_{6\rho_{5}}\sim\cdot\sim$ . Sowe have the coset decomposition $\mathcal{Z}\supset \mathcal{Z}_{l}\cup Z_{l}\cdot\rho_{1^{-}}\cup \mathcal{Z}_{l}\cdot\rho_{2}\sim\cup Z_{l}\cdot\rho 5^{-}$. And the estimatesof the global indices of both hand side is $|\mathrm{A}_{29}|=(2\beta^{2}-4\beta+4)|E_{8}|$ and $|E_{8}|+\beta|E_{8}|$

$+(\beta-1)^{2}|E_{8}|+\beta(-\beta^{3}+7\beta^{2}-13\beta+5)^{2}|E_{8}|=(2\beta^{2}-4\beta+4)|E_{8}|$ , where we used theequality $\beta^{4}-7\beta^{3}+14\beta^{2}-8\beta+1=0$ to colnpute the right hand side. So we get the equality

131

evenThe chiral left part

oddeven

The chiral right part

odd

Figure 38: The chiral left and right part of the fusion rule algebra $Z$ of $E_{8}$ .

Table 2: Multiplication table for $\cdot\otimes_{N}\cdot$ of the fusion rule algebra of $E_{8}(1)$

in the above coset decomposition. This shows that the whole system $Z$ is generated bythe two elements $p_{1}^{+}$ and $p_{1}^{-}$ . Finally by looking at the index values and the fusion rule,we obtain the fusion rule graph for the two generators $[1]=p_{1}^{+}$ and $[1^{\sim}]=p_{1}^{-}$ as in inFigure 45.

Remark 5.12 We will explain the meaning of Figures 40 to 45. The white verticesand black vertices represents even and odd vertices $re\mathit{8}pectively$ . The large double circledvertices denote the ambichiral part. The thick edges and thin edges represent the chiral leftgraphs and the left coset graphs, which are obtained as Cayley graphs for multiplicationby the generator $p_{1}^{+}$ from the left. The thick dotted edges and thin dotted $edge\mathit{8}$ representthe chiral right graphs and the right coset $graph_{\mathit{8}}$ , which are obtained as Cayley graphs for

$N-N$

$N-M$

$M-N$

$M-M$

Figure 39: The (dual) principal graph $E_{8}$

132

Table 3: Multiplication table for $\cdot\otimes_{N}\cdot$ of. the fusion rule algebra of $E_{8}(2)$

multiplication by the generator $p_{1}^{-}$ from the right.

In the procedure to get the complete classification of irreducible K-K $\mathrm{b}\mathrm{i}$-unitary con-nections. We also obtained the complete classification of fiat connections and flat part ofnon-flat connections on the Dynkin diagrams. We state this as the following corollary.

Corollary 5.13 The (fundamental) $bi$-unitary connections on the four graphs as in Fig-ure 35 are flat in the case of $A_{n},$ $D_{2n},$ $E_{6}$ and $E_{8}$ . They are not flat in the case of $D_{2n+1}$

and $E_{7}$ . The flat part of $D_{2n+1}$ and $E_{\overline{\prime}}$ connections are $A_{4n-1}$ and $D_{10}$ respectively.

Hence it provides another proof of the colnplete classification of subfactors of thehyperfinite $\mathrm{I}\mathrm{I}_{1}$ factor with index less than 4.

Corollary 5.14 There is only one subfactor with principal graph $A_{n}$ for each $n\geq 2$ .There is only one subfactor with principal graph $D_{2n}$ for each $n\geq 2$ . There are two andonly two $non- i_{\mathit{8}}omorphiC$ subfactors with principal graph $E_{6}$ and $E_{8}$ respectively. Theseare all the subfactors of the hyperfinite $II_{1}$ factor with index less than 4.

By examining each case we obtain the following structural result on the fusion rulealgebra of all K-K $\mathrm{b}\mathrm{i}$-unitary connections.

Theorem 5.15 Let $K$ be one of the A-D-E Dynkin diagrams. The fusion rule algebrasof all K-K $bi$-unitary connections are generated by the two minimal central projections

$p_{1}^{\pm}$ . Moreover the chiral left part and the chiral right part commutes.

Remark 5.16 The first assertion of the above theorem can be shown directly by takingthe . product of two chiral projectors $\Psi_{+}\cdot\Psi_{-}$ . lVIinimal central projections appear in thisproduct is contained in the $fu\mathit{8}i_{on}$ rule subalgebra generated by $p_{1}^{+}$ and $p_{1}^{-}$ . And it is notdifficult to show that the product contains the identity element of the K-K double trianglealgebra by using the non-degenerate braiding on the recoupling $\mathit{8}ystem$ A. This result canbe generalized in more abstract $\mathit{8}etting$ of

$\cdot$

double triangle algebra as in [5].

Remark 5.17 Though it is not written in detail in [24], Ocneanu showed stronger resultthan the commutativity of the chiral left and nght part. He showed that the chiral leftpart and chiral right part of the fusion rule algebra of K-K $bi$-unitary connections has

133

$[0][egg0]$

$A_{1}$ $A_{2}$ $A_{3}$ $A_{4}$ $A_{5}$

Figure 40: Chiral symlnetry for the Coxeter graph $A_{n}$

non-degenerated braiding. (See the explanation of the picture $‘(Quantum$ Symmetry forCoxeter graphs” in [24].) He defined the choice of intertwiner in $\mathrm{H}\mathrm{o}\mathrm{m}(p_{i}\cdot p_{j},p_{j}\cdot p_{1})$

graphically and $\mathit{8}howed$ the $exi\mathit{8}tence$ of the non-degenerate braiding. This shows that thefusion rule algebra of the ambichiral part has non-degenerate braiding. And it impliesthe existence of non-degenerate braiding on the $s$ ystern of bimodule corresponding to evenvertices of the Dynkin diagram $D_{2n}u$)$hich$ was shown by D. E. Evans-Y. Kawahigashi in[12]. .

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Figure 41: Chiral symnletry for the Coxeter graph $D_{odd}$

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$D_{4}$ $D_{6}$ $D_{8}$

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Figure 43: Chiral symnuetry for the Coxeter graph $E_{6}$

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Figure 44: Chiral $\mathrm{s}\mathrm{y}_{\mathrm{l}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{t}\Gamma}\mathrm{y}$ for the Coxeter graph $E_{7}$

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$\lceil 01$

Figure 45: Chiral symlnetry for the Coxeter graph $E_{8}$

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